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Nonexistence of universal orders in many cardinals

1992-09-01
Menachem Kojman , Saharon Shelah
Abstract Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which … Abstract Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ 1 ; without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove—again in ZFC—that for a large class of cardinals there is no universal linear order (e.g. in every regular ). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles” ℵ 1 —a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p -adic rings and fields, partial orders, models of PA and so on).
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Convex decompositions in the plane and continuous pair colorings of the irrationals

2002-12-01
Stefan Geschke , Menachem Kojman , Wiesław Kubiś +1 more

Infinite homogeneous bipartite graphs with unequal sides

1996-02-01
Martin Goldstern , Rami Grossberg , Menachem Kojman
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The universality spectrum of stable unsuperstable theories

1992-07-01
Menachem Kojman , Saharon Shelah
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A ZFC Dowker space in ℵ_{𝜔+1}: An application of pcf theory to topology

1998-01-01
Menachem Kojman , Saharon Shelah
The existence of a Dowker space of cardinality<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript omega plus 1"><mml:semantics><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ω</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\aleph _{\omega +1}</mml:annotation></mml:semantics></mml:math></inline-formula>and weight<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript omega plus 1"><mml:semantics><mml:msub><mml:mi … The existence of a Dowker space of cardinality<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript omega plus 1"><mml:semantics><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ω</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\aleph _{\omega +1}</mml:annotation></mml:semantics></mml:math></inline-formula>and weight<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript omega plus 1"><mml:semantics><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ω</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\aleph _{\omega +1}</mml:annotation></mml:semantics></mml:math></inline-formula>is proved in ZFC using pcf theory.
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Universal abelian groups

1995-02-01
Menachem Kojman , Saharon Shelah
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Van der Waerden spaces

2001-08-28
Menachem Kojman
A topological space $X$ is van der Waerden if for every sequence $(x_n)_n$ in $X$ there exists a converging subsequence $(x_{n_k})_k$ so that $\{{n_k}:k\in \mathbb {N}\}$ contains arbitrarily long finite … A topological space $X$ is van der Waerden if for every sequence $(x_n)_n$ in $X$ there exists a converging subsequence $(x_{n_k})_k$ so that $\{{n_k}:k\in \mathbb {N}\}$ contains arbitrarily long finite arithmetic progressions. Not every sequentially compact space is van der Waerden. The product of two van der Waerden spaces is van der Waerden. The following condition on a Hausdorff space $X$ is sufficent for $X$ to be van der Waerden: [$(*)$] The closure of every countable set in $X$ is compact and first-countable. A Hausdorff space $X$ that satisfies $(*)$ satisfies, in fact, a stronger property: for every sequence $(x_n)$ in $X$: [$(\star )$] There exists $A\subseteq \mathbb {N}$ so that $(x_n)_{n\in A}$ is converging, and $A$ contains arbitrarily long finite arithmetic progressions and sets of the form $FS(D)$ for arbitrarily large finite sets $D$. There are nonmetrizable and noncompact spaces which satisfy $(*)$. In particular, every sequence of ordinal numbers and every bounded sequence of real monotone functions on $[0,1]$ satisfy $(\star )$.
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CONTINUOUS RAMSEY THEORY ON POLISH SPACES AND COVERING THE PLANE BY FUNCTIONS

2004-12-01
Stefan Geschke , Martin Goldstern , Menachem Kojman
We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: … We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: see text] of a pair-coloring c:[X] 2 →2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2 ω , c min and c max , which satisfy [Formula: see text] and prove: Theorem. (1) For every Polish space X and every continuous pair-coloringc:[X] 2 →2with[Formula: see text], [Formula: see text] (2) There is a model of set theory in which[Formula: see text]and[Formula: see text]. The consistency of [Formula: see text] and of [Formula: see text] follows from [20]. We prove that [Formula: see text] is equal to the covering number of (2 ω ) 2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c min gives: Theorem. There is a model of set theory in which (1) ℝ 2 is coverable byℵ 1 graphs and reflections of graphs of continuous real functions; (2) ℝ 2 is not coverable byℵ 1 graphs and reflections of graphs of Lipschitz real functions. Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes [Formula: see text] from row (3).
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Representing Embeddability as Set Inclusion

1998-10-01
Menachem Kojman
A few steps are made towards a representation theory of embeddability among uncountable graphs. A class of graphs is defined by forbidding some countable configurations which are related to the … A few steps are made towards a representation theory of embeddability among uncountable graphs. A class of graphs is defined by forbidding some countable configurations which are related to the graph's end-structure. Using uncountable combinatorics, a representation theorem for embeddability in this class is proved, which asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality λ>ℵ1 onto set inclusion over all subsets of reals or cardinality λ or less. As corollaries we obtain: (1) the complexity of the class in every regular uncountable λ>ℵ1 is at least λ + + sup { μ ℵ 0 : μ + < λ } , (2) a characterisation of graphs in the class for which some invariant of the graph has to be inherited by one of fewer than λ subgraphs whose union covers G. Some continuity properties of the homomorphism in the representation theorem are explored and are used to extend the first corollary to all singular cardinals below the first fixed point of second order. The first corollary shows that, contrary to what Shelah has shown for the class of all graphs, the relations of embeddability in the class under discussion is not independent of negations of the generalised continuum hypothesis.
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Sharp thresholds for the phase transition between primitive recursive and Ackermannian Ramsey numbers

2008-03-11
Menachem Kojman , Gyesik Lee , Eran Omri +1 more

Exact upper bounds and their uses in set theory

1998-08-01
Menachem Kojman

Noetherian type in topological products

2014-07-01
Menachem Kojman , David Milovich , Santi Spadaro
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Sets in a euclidean space which are not a countable union of convex subsets

1990-10-01
Menachem Kojman , Micha A. Perles , Saharon Shelah

Regressive Ramsey Numbers Are Ackermannian

1999-04-01
Menachem Kojman , Saharon Shelah
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Hindman spaces

2001-12-20
Menachem Kojman
A topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>Hindman</italic> if for every sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n … A topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>Hindman</italic> if for every sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists an infinite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D subset-of-or-equal-to double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊆</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">D\subseteq \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that the sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n element-of upper F upper S left-parenthesis upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>F</mml:mi> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_{n\in FS(D)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, indexed by all finite sums over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is IP-converging in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Not all sequentially compact spaces are Hindman. The product of two Hindman spaces is Hindman. Furstenberg and Weiss proved that all compact metric spaces are Hindman. We show that every Hausdorff space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that satisfies the following condition is Hindman: <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis reverse-solidus quad The closure of every countable set in upper X is compact and first hyphen countable period reverse-solidus quad"> <mml:semantics> <mml:mrow> <mml:mtext>(</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mtext>)\quad The closure of every countable set in </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi> </mml:mrow> <mml:mtext> is compact and first-countable.\quad </mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\text {($*$)\quad The closure of every countable set in $X$ is compact and first-countable.\quad }</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> Consequently, there exist nonmetrizable and noncompact Hindman spaces. The following is a particular consequence of the main result: every bounded sequence of monotone (not necessarily continuous) real functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has an IP-converging subsequences.

Universal arrow-free graphs

1996-01-01
Martin Goldstern , Menachem Kojman

?-complete Souslin trees on ?+

1993-05-01
Menachem Kojman , Saharon Shelah

Convexity ranks in higher dimensions

2000-01-01
Menachem Kojman
A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed … A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure
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Splitting families of sets in ZFC

2014-11-13
Menachem Kojman
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Van der Waerden spaces and Hindman spaces are not the same

2002-12-16
Menachem Kojman , Saharon Shelah
A Hausdorff topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>van der Waerden</italic> if for every sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis … A Hausdorff topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>van der Waerden</italic> if for every sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n element-of omega"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_{n\in \omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there is a converging subsequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n element-of upper A"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_{n\in A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A subset-of-or-equal-to omega"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A\subseteq \omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains arithmetic progressions of all finite lengths. A Hausdorff topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>Hindman</italic> if for every sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n element-of omega"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_{n\in \omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there is an <italic>IP-converging</italic> subsequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n element-of upper F upper S left-parenthesis upper B right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>F</mml:mi> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_{n\in FS(B)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some infinite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B subset-of-or-equal-to omega"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">B\subseteq \omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.

Almost isometric embedding between metric spaces

2006-12-01
Menachem Kojman , Saharon Shelah

Cantor-Bendixson degrees and convexity in ℝ2

2001-12-01
Menachem Kojman

Shelah’s revised GCH theorem and a question by Alon on infinite graphs colorings

2014-02-07
Menachem Kojman

On the arithmetic of density

2016-08-23
Menachem Kojman

STRONG COLORINGS OVER PARTITIONS

2021-02-22
William Chen-Mertens , Menachem Kojman , Juris Steprāns
A strong coloring on a cardinal $\kappa$ is a function $f:[\kappa]^2\to \kappa$ such that for every $A\subseteq \kappa$ of full size $\kappa$, every color $\gamma<\kappa$ is attained by $f\upharpoonright[A]^2$. The … A strong coloring on a cardinal $\kappa$ is a function $f:[\kappa]^2\to \kappa$ such that for every $A\subseteq \kappa$ of full size $\kappa$, every color $\gamma<\kappa$ is attained by $f\upharpoonright[A]^2$. The symbol $\kappa\nrightarrow [\kappa]^2_\kappa$ asserts the existence of a strong coloring on $\kappa$. We introduce the symbol $\kappa\nrightarrow_p[\kappa]^2_\kappa$ which asserts the existence of a coloring $f:[\kappa]^2\to \kappa$ which is strong over a partition $p:[\kappa]^2\to\theta$. A coloring $f$ is strong over $p$ if for every $A\in [\kappa]^\kappa$ there is $i<\theta$ so that every color $\gamma<\kappa$ is attained by $f\upharpoonright ([A]^2\cap p^{-1}(i))$. We prove that whenever $\kappa\nrightarrow[\kappa]^2_\kappa$ holds, also $\kappa\nrightarrow_p[\kappa]^2_\kappa$ holds for an arbitrary finite partition $p$. Similarly, arbitrary finite $p$-s can be added to stronger symbols which hold in any model of ZFC. If $\kappa^\theta=\kappa$, then $\kappa\nrightarrow_p[\kappa]^2_\kappa$ and stronger symbols, like $\mathrm{Pr}_1(\kappa,\kappa,\kappa,\chi)$ or $\mathrm{Pr}_0(\kappa,\kappa,\kappa,\aleph_0)$, hold also for an arbitrary partition $p$ to $\theta$ parts.
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Sequentially linearly Lindelöf spaces

2003-02-01
Menachem Kojman , Victoria Lubitch

Homogeneous Families and their Automorphism Groups

1995-10-01
Menachem Kojman , Saharon Shelah
A homogeneous family of subsets over a given set is one with a very ‘rich’ automorphism group. We prove the existence of bi-universal element in the class of homogeneous families … A homogeneous family of subsets over a given set is one with a very ‘rich’ automorphism group. We prove the existence of bi-universal element in the class of homogeneous families over a given infinite set and give an explicit construction of 2 2 ℵ 0 isomorphism types of homogeneous families over a countable set.
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On two problems of Erdos and Hechler: New methods in singular madness

2004-06-21
Menachem Kojman , Wiesław Kubiś , Saharon Shelah
For an infinite cardinal $\mu$, $\operatorname {MAD}(\mu )$ denotes the set of all cardinalities of nontrivial maximal almost disjoint families over $\mu$. Erdős and Hechler proved in 1973 the consistency … For an infinite cardinal $\mu$, $\operatorname {MAD}(\mu )$ denotes the set of all cardinalities of nontrivial maximal almost disjoint families over $\mu$. Erdős and Hechler proved in 1973 the consistency of $\mu \in \operatorname {MAD}(\mu )$ for a singular cardinal $\mu$ and asked if it was ever possible for a singular $\mu$ that $\mu \notin \operatorname {MAD}(\mu )$, and also whether $2^{\operatorname {cf}\mu } <\mu \Longrightarrow \mu \in \operatorname {MAD}(\mu )$ for every singular cardinal $\mu$. We introduce a new method for controlling $\operatorname {MAD} (\mu )$ for a singular $\mu$ and, among other new results about the structure of $\operatorname {MAD}(\mu )$ for singular $\mu$, settle both problems affirmatively.
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Singular Cardinals

2012-01-01
Menachem Kojman

The A,B,C of pcf: a companion to pcf theory, Part I

1995-01-01
Menachem Kojman
This is a survey paper of Shelah's pcf theory. In this part the theory is developed up to the pcf theorem. This is a survey paper of Shelah's pcf theory. In this part the theory is developed up to the pcf theorem.

Fallen Cardinals

2000-09-07
Menachem Kojman , Saharon Shelah
We prove that for every singular cardinal mu of cofinality omega, the complete Boolean algebra compP_mu(mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra Comp Col(omega_1,mu^{aleph_0}). … We prove that for every singular cardinal mu of cofinality omega, the complete Boolean algebra compP_mu(mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra Comp Col(omega_1,mu^{aleph_0}). Consequently, adding a generic filter to the quotient algebra P_mu(mu)=P(mu)/[mu]^{<mu} collapses mu^{aleph_0} to aleph_1. Another corollary is that the Baire number of the space U(mu) of all uniform ultrafilters over mu is equal to omega_2. The corollaries affirm two conjectures by Balcar and Simon. The proof uses pcf theory.

Fallen cardinals

2001-05-01
Menachem Kojman , Saharon Shelah

Ramsey theory over partitions III: Strongly Luzin sets and partition relations

2022-04-27
Menachem Kojman , Assaf Rinot , Juris Steprāns
The strongest type of coloring of pairs of countable ordinals, gotten by Todorčević from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set … The strongest type of coloring of pairs of countable ordinals, gotten by Todorčević from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\aleph _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it. This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpiński a hundred years ago.

Symmetrized Induced Ramsey Theory

2011-01-04
Stefan Geschke , Menachem Kojman

Advances on strong colorings over partitions

2021-02-14
Menachem Kojman , Assaf Rinot , Juris Steprāns
We advance the theory of strong colorings over partitions, studying both positive and negative Ramsey relations at the level of the uncountable. A correspondence between combinatorial properties of partitions and … We advance the theory of strong colorings over partitions, studying both positive and negative Ramsey relations at the level of the uncountable. A correspondence between combinatorial properties of partitions and chain conditions of natural forcing notions for destroying strong colorings over them is uncovered and enables us to prove positive Ramsey relations for $\aleph_1$ from weak forms of Martin's Axiom, thereby answering two questions from [CKS21]. Positive Ramsey relations for $\aleph_2$ and higher cardinals are established as well and without making use of large cardinals. We also provide a group of pump-up theorems for strong colorings over partitions. Some of them solve more problems from [CKS21].

Order-theoretic properties of bases in topological spaces I

2010-12-17
Menachem Kojman , David Milovich , Santi Spadaro
The cardinal invariant of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as … The cardinal invariant of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_\omega$ with the countable box topology, $(2^{\aleph_\omega})_\delta$, is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of $\aleph_\omega$. We discuss the influence of principles like $\square_{\aleph_\omega}$ and Chang's conjecture for $\aleph_\omega$ on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms). Within PCF theory we establish the existence of an $(\aleph_4,\aleph_1)$-sparse covering family of countable subsets of $\aleph_\omega$. From this follows an absolute upper bound of $\aleph_4$ on the Noetherian type of $(2^{\aleph_\omega})_\delta$. The proof uses ideas from Shelah's proof that if $\kappa^+ <\lambda$ then his ideal $I[\lambda]$ contains a stationary set consisting of points of cofinality $\kappa$.

On universal graphs without cliques or withour large bipartite graphs

1995-07-18
Menachem Kojman
For every uncountable cardinal $\lambda$, suitable negations of the Generalized Continuum Hypothesis imply: - For all infinite $\alpha$ and $\beta$, there is no universal $K_{\alpha,\beta}$-free graphs in $\lambda$ - For … For every uncountable cardinal $\lambda$, suitable negations of the Generalized Continuum Hypothesis imply: - For all infinite $\alpha$ and $\beta$, there is no universal $K_{\alpha,\beta}$-free graphs in $\lambda$ - For all $\alpha\ge 3$, there is no universal $K_\alpha$-free graph in $\lambda$ The instance $K_{\omega,\omega_1}$ for $\lambda=\aleph_1$ was settled by Komjath and Pach from the principle $\diamondsuit(\omega_1)$.

Rules and reals

1999-01-29
Martin Goldstern , Menachem Kojman
A “<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rule" is a sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With right-arrow equals left-parenthesis left-parenthesis upper A … A “<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rule" is a sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With right-arrow equals left-parenthesis left-parenthesis upper A Subscript n Baseline comma upper B Subscript n Baseline right-parenthesis colon n greater-than double-struck upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\vec A=((A_n,B_n): n&gt;\mathbb N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of pairwise disjoint sets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript n"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">B_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, each of cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to k"> <mml:semantics> <mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\le k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and subsets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript n Baseline subset-of-or-equal-to upper B Subscript n"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>⊆</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">A_n\subseteq B_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X subset-of-or-equal-to double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊆</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">X\subseteq \mathbb N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (a “real”) follows a rule <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With right-arrow"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\vec A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if for infinitely many <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">n\in \mathbb N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X intersection upper B Subscript n Baseline equals upper A Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>∩</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">X\cap B_n=A_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rules, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German s Subscript k"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {s}_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the least number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rules with no real that follows all of them, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German r Subscript k"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {r}_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Call <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With right-arrow"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\vec A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a <italic>bounded</italic> rule if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With right-arrow"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\vec A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rule for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German r Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {r}_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the least cardinality of a set of bounded rules with no real following all rules in the set. We prove the following: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German r Subscript normal infinity Baseline greater-than-or-equal-to max left-parenthesis c o v left-parenthesis double-struck upper K right-parenthesis comma c o v left-parenthesis double-struck upper L right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msub> <mml:mo>≥</mml:mo> <mml:mo movablelimits="true" form="prefix">max</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>cov</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>cov</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {r}_\infty \ge \max (\operatorname {cov}(\mathbb {K}),\operatorname {cov}(\mathbb {L}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German r equals German r 1 greater-than-or-equal-to German r 2 equals German r Subscript k"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≥</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {r}=\mathfrak {r}_1\ge \mathfrak {r}_2=\mathfrak {r}_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k\ge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. However, in the Laver model, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German r 2 greater-than German b equals German r 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">b</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {r}_2&gt;\mathfrak {b}=\mathfrak {r}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. An application of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German r Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {r}_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is in Section 3: we show that below <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German r Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">r</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {r}_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> one can find proper extensions of dense independent families which preserve a pre-assigned group of automorphisms. The original motivation for discovering rules was an attempt to construct a maximal homogeneous family over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The consistency of such a family is still open.

Countably convex G<sub>δ</sub>sets

2001-01-01
V. P. Fonf , Menachem Kojman

Partitioning Subgraphs of Profinite Ordered Graphs

2018-08-14
Stefanie Huber , Stefan Geschke , Menachem Kojman

A proof of Shelah's partition theorem

1995-10-01
Menachem Kojman

On symmetric indivisbility of countable structures

2011-01-01
Assaf Hasson , Menachem Kojman , Alf Onshuus

A proof of Shelah's partition theorem

1995-10-01
Menachem Kojman

A symmetrized metric ramsey theorem

2009-09-01
Menachem Kojman

Metric Baumgartner theorems and universality

2007-01-01
Stefan Geschke , Menachem Kojman
It is consistent with the axioms of set theory that for every metric space X which is isometric to some separable Banach space or to Urysohn's universal separable metric space … It is consistent with the axioms of set theory that for every metric space X which is isometric to some separable Banach space or to Urysohn's universal separable metric space U the following holds:( ) X There exists a nowhere meager subspace of X of cardinality ℵ 1 and any two nowhere meager subsets of X of cardinality ℵ 1 are almost isometric to each other.As a corollary, it is consistent that the Continuum Hypothesis fails and the following hold:(1) There exists an almost-isometry ultrahomogeneous and universal element in the class of separable metric spaces of size ℵ 1 .(2) For every separable Banach space X there exists an almost-isometry conditionally ultrahomogeneous and universal element in the class of subspaces of X of size ℵ 1 .(3) For every finite dimensional Banach space X, there is a unique universal element up to almost-isometry in the class of subspaces of X of size ℵ 1
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Almost isometric embeddings of metric spaces

2004-01-01
Menachem Kojman , Saharon Shelah
We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. … We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum separable metric spaces on aleph_1 so that every separable metric space is almost isometrically embedded into one of them when the continuum hypothesis fails. (3) There is no collection of fewer than continuum metric spaces of cardinality aleph_2 so that every ultra-metric space of cardinality aleph_2 is almost isometrically embedded into one of them if aleph_2&lt;2^{aleph_0}. We also prove that various spaces X satisfy that if a space X is almost isometric to X than Y is isometric to X.

Noetherian type in topological products

2010-12-17
Menachem Kojman , David Milovich , Santi Spadaro
The cardinal invariant of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as … The cardinal invariant of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_\omega$ with the countable box topology, $(2^{\aleph_\omega})_\delta$, is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of $\aleph_\omega$. We discuss the influence of principles like $\square_{\aleph_\omega}$ and Chang's conjecture for $\aleph_\omega$ on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms). Within PCF theory we establish the existence of an $(\aleph_4,\aleph_1)$-sparse covering family of countable subsets of $\aleph_\omega$. From this follows an absolute upper bound of $\aleph_4$ on the Noetherian type of $(2^{\aleph_\omega})_\delta$. The proof uses ideas from Shelah's proof that if $\kappa^+ <\lambda$ then his ideal $I[\lambda]$ contains a stationary set consisting of points of cofinality $\kappa$.

PCF Theory

2005-01-19
Menachem Kojman

Sierpinski's onto mapping and partitions

2021-04-19
Menachem Kojman , Assaf Rinot , Juris Steprāns
In his 1987 paper, Todorcevic remarks that Sierpinski's onto mapping principle (1932) and the Erdos-Hajnal-Milner negative Ramsey relation (1966) are equivalent to each other, and follow from the existence of … In his 1987 paper, Todorcevic remarks that Sierpinski's onto mapping principle (1932) and the Erdos-Hajnal-Milner negative Ramsey relation (1966) are equivalent to each other, and follow from the existence of a Luzin set. Recently, Guzman and Miller showed that these two principles are also equivalent to the existence of a nonmeager set of reals of cardinality $\aleph_1$. We expand this circle of equivalences and show that these propositions are equivalent also to the high-dimensional version of the Erdos-Hajnal-Milner negative Ramsey relation, thereby improving a CH theorem of Galvin (1980). Then we consider the validity of these relations in the context of strong colorings over partitions and prove the consistency of a positive Ramsey relation, as follows: It is consistent with the existence of both a Luzin set and of a Souslin tree that for some countable partition p, all colorings are p-special.

Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems

2023-11-13
Menachem Kojman , Assaf Rinot , Juris Steprāns

Ramsey theory over partitions I: Positive Ramsey relations from forcing axioms

2023-11-13
Menachem Kojman , Assaf Rinot , Juris Steprāns

Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems

2023-11-13
Menachem Kojman , Assaf Rinot , Juris Steprāns

Ramsey theory over partitions III: Strongly Luzin sets and partition relations

2022-04-27
Menachem Kojman , Assaf Rinot , Juris Steprāns
The strongest type of coloring of pairs of countable ordinals, gotten by Todorčević from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set … The strongest type of coloring of pairs of countable ordinals, gotten by Todorčević from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\aleph _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it. This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpiński a hundred years ago.

Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems

2022-01-01
Menachem Kojman , Assaf Rinot , Juris Steprāns
In this series of papers we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and … In this series of papers we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in particular solve two problems from [CKS21]. It is shown that for every infinite cardinal $\lambda$, a strong coloring on $\lambda^+$ by $\lambda$ colors over a partition can be stretched to one with $\lambda^{+}$ colors over the same partition. Also, a sufficient condition is given for when a strong coloring witnessing $Pr_1(\ldots)$ over a partition may be improved to witness $Pr_0(\ldots)$. Since the classical theory corresponds to the special case of a partition with just one cell, the two results generalize pump-up theorems due to Eisworth and Shelah, respectively.
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Sierpinski's onto mapping and partitions

2021-04-19
Menachem Kojman , Assaf Rinot , Juris Steprāns
In his 1987 paper, Todorcevic remarks that Sierpinski's onto mapping principle (1932) and the Erdos-Hajnal-Milner negative Ramsey relation (1966) are equivalent to each other, and follow from the existence of … In his 1987 paper, Todorcevic remarks that Sierpinski's onto mapping principle (1932) and the Erdos-Hajnal-Milner negative Ramsey relation (1966) are equivalent to each other, and follow from the existence of a Luzin set. Recently, Guzman and Miller showed that these two principles are also equivalent to the existence of a nonmeager set of reals of cardinality $\aleph_1$. We expand this circle of equivalences and show that these propositions are equivalent also to the high-dimensional version of the Erdos-Hajnal-Milner negative Ramsey relation, thereby improving a CH theorem of Galvin (1980). Then we consider the validity of these relations in the context of strong colorings over partitions and prove the consistency of a positive Ramsey relation, as follows: It is consistent with the existence of both a Luzin set and of a Souslin tree that for some countable partition p, all colorings are p-special.

STRONG COLORINGS OVER PARTITIONS

2021-02-22
William Chen-Mertens , Menachem Kojman , Juris Steprāns
A strong coloring on a cardinal $\kappa$ is a function $f:[\kappa]^2\to \kappa$ such that for every $A\subseteq \kappa$ of full size $\kappa$, every color $\gamma<\kappa$ is attained by $f\upharpoonright[A]^2$. The … A strong coloring on a cardinal $\kappa$ is a function $f:[\kappa]^2\to \kappa$ such that for every $A\subseteq \kappa$ of full size $\kappa$, every color $\gamma<\kappa$ is attained by $f\upharpoonright[A]^2$. The symbol $\kappa\nrightarrow [\kappa]^2_\kappa$ asserts the existence of a strong coloring on $\kappa$. We introduce the symbol $\kappa\nrightarrow_p[\kappa]^2_\kappa$ which asserts the existence of a coloring $f:[\kappa]^2\to \kappa$ which is strong over a partition $p:[\kappa]^2\to\theta$. A coloring $f$ is strong over $p$ if for every $A\in [\kappa]^\kappa$ there is $i<\theta$ so that every color $\gamma<\kappa$ is attained by $f\upharpoonright ([A]^2\cap p^{-1}(i))$. We prove that whenever $\kappa\nrightarrow[\kappa]^2_\kappa$ holds, also $\kappa\nrightarrow_p[\kappa]^2_\kappa$ holds for an arbitrary finite partition $p$. Similarly, arbitrary finite $p$-s can be added to stronger symbols which hold in any model of ZFC. If $\kappa^\theta=\kappa$, then $\kappa\nrightarrow_p[\kappa]^2_\kappa$ and stronger symbols, like $\mathrm{Pr}_1(\kappa,\kappa,\kappa,\chi)$ or $\mathrm{Pr}_0(\kappa,\kappa,\kappa,\aleph_0)$, hold also for an arbitrary partition $p$ to $\theta$ parts.
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Advances on strong colorings over partitions

2021-02-14
Menachem Kojman , Assaf Rinot , Juris Steprāns
We advance the theory of strong colorings over partitions, studying both positive and negative Ramsey relations at the level of the uncountable. A correspondence between combinatorial properties of partitions and … We advance the theory of strong colorings over partitions, studying both positive and negative Ramsey relations at the level of the uncountable. A correspondence between combinatorial properties of partitions and chain conditions of natural forcing notions for destroying strong colorings over them is uncovered and enables us to prove positive Ramsey relations for $\aleph_1$ from weak forms of Martin's Axiom, thereby answering two questions from [CKS21]. Positive Ramsey relations for $\aleph_2$ and higher cardinals are established as well and without making use of large cardinals. We also provide a group of pump-up theorems for strong colorings over partitions. Some of them solve more problems from [CKS21].

Ramsey theory over partitions III: Strongly Luzin sets and partition relations

2021-01-01
Menachem Kojman , Assaf Rinot , Juris Steprāns
The strongest type of coloring of pairs of countable ordinals, gotten by Todorcevic from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set … The strongest type of coloring of pairs of countable ordinals, gotten by Todorcevic from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size $\aleph_1$. In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it. This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpinski a hundred years ago.
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Ramsey theory over partitions I: Positive Ramsey relations from forcing axioms

2021-01-01
Menachem Kojman , Assaf Rinot , Juris Steprāns
In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing … In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over them is uncovered. At the level of the first uncountable cardinal this gives rise to a duality theorem under Martin's Axiom: a function $p:[\omega_1]^2\rightarrow\omega$ witnesses a weak negative Ramsey relation when $p$ plays the role of a coloring if and only if a positive Ramsey relation holds over $p$ when $p$ plays the role of a partition. The consistency of positive Ramsey relations over partitions does not stop at the first uncountable cardinal: it is established that at any prescribed uncountable cardinal these relations follow from forcing axioms without large cardinal strength. This result solves in particular two problems from [CKS21].
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Partitioning Subgraphs of Profinite Ordered Graphs

2018-08-14
Stefanie Huber , Stefan Geschke , Menachem Kojman

On the arithmetic of density

2016-08-23
Menachem Kojman

On the arithmetic of density

2015-10-08
Menachem Kojman
The $\kappa$-density of a cardinal $\mu\ge\kappa$ is the least cardinality of a dense collection of $\kappa$-subsets of $\mu$ and is denoted by $\mathcal D(\mu,\kappa)$. The Singular Density Hypothesis (SDH) for … The $\kappa$-density of a cardinal $\mu\ge\kappa$ is the least cardinality of a dense collection of $\kappa$-subsets of $\mu$ and is denoted by $\mathcal D(\mu,\kappa)$. The Singular Density Hypothesis (SDH) for a singular cardinal $\mu$ of cofinality $cf\mu=\kappa$ is the equation $\mathcal D(\mu,\kappa)=\mu^+$. The Generalized Density Hypothesis (GDH) for $\mu$ and $\lambda$ such that $\lambda\le\mu$ is: $\mathcal D(\mu,\lambda)=\mu$ if $cf\mu\not=cf\lambda$ and $\mathcal D(\mu,\lambda)=\mu^+$ if $cf\mu=cf\lambda$. Density is shown to satisfy Silver's theorem. The most important case is: Theorem 2.6. If $\kappa=cf\kappa<\theta=cf\mu<\mu$ and the set of cardinals $\lambda<\mu$ of cofinality $\kappa$ that satisfy the \textsf{SDH} is stationary in $\mu$ then the SDH holds at $\mu$. A more general version is given in Theorem 2.8 A corollary of Theorem 2.6 is: Theorem 3.2 If the Singular Density Hypothesis holds for all sufficiently large singular cardinals of some fixed cofinality $\kappa$, then for all cardinals $\lambda$ with $cf\lambda \ge \kappa$, for all sufficiently large $\mu$, the GDH holds.

On the arithmetic of density

2015-01-01
Menachem Kojman
The $κ$-density of a cardinal $μ\geκ$ is the least cardinality of a dense collection of $κ$-subsets of $μ$ and is denoted by $\mathcal D(μ,κ)$. The Singular Density Hypothesis (SDH) for … The $κ$-density of a cardinal $μ\geκ$ is the least cardinality of a dense collection of $κ$-subsets of $μ$ and is denoted by $\mathcal D(μ,κ)$. The Singular Density Hypothesis (SDH) for a singular cardinal $μ$ of cofinality $cfμ=κ$ is the equation $\mathcal D(μ,κ)=μ^+$. The Generalized Density Hypothesis (GDH) for $μ$ and $λ$ such that $λ\leμ$ is: $\mathcal D(μ,λ)=μ$ if $cfμ\not=cfλ$ and $\mathcal D(μ,λ)=μ^+$ if $cfμ=cfλ$. Density is shown to satisfy Silver's theorem. The most important case is: Theorem 2.6. If $κ=cfκ

Splitting families of sets in ZFC

2014-11-13
Menachem Kojman
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Noetherian type in topological products

2014-07-01
Menachem Kojman , David Milovich , Santi Spadaro
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Shelah’s revised GCH theorem and a question by Alon on infinite graphs colorings

2014-02-07
Menachem Kojman

Splitting families of sets in ZFC

2012-09-06
Menachem Kojman
Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,\rho)$ in which $n$ is finite and $\rho$ is infinite. An extension of Miller's theorem is proved here in ZFC … Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,\rho)$ in which $n$ is finite and $\rho$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(\nu,\rho)$ in which $\nu$ is arbitrary and $\rho\ge \beth_\om(\nu)$. The proof uses a new general method that is based on Shelah's revises Generalized Continuum Hypothesis theorem. Upper bounds on conflict-free coloring numbers of families of sets and a general comparison theorem follow as corollaries of the main theorem. Other corollaries eliminate the use of additional axioms from splitting theorems due to Erdos, Hajnal, Komjath, Juhasz and Shelah.

Singular Cardinals

2012-01-01
Menachem Kojman

Splitting families of sets in ZFC

2012-01-01
Menachem Kojman
Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,\rho)$ in which $n$ is finite and $\rho$ is infinite. An extension of Miller's theorem is proved here in ZFC … Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,\rho)$ in which $n$ is finite and $\rho$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(\nu,\rho)$ in which $\nu$ is arbitrary and $\rho\ge \beth_\om(\nu)$. The proof uses a new general method that is based on Shelah's revises Generalized Continuum Hypothesis theorem. Upper bounds on conflict-free coloring numbers of families of sets and a general comparison theorem follow as corollaries of the main theorem. Other corollaries eliminate the use of additional axioms from splitting theorems due to Erdos, Hajnal, Komjath, Juhasz and Shelah.
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Symmetrized Induced Ramsey Theory

2011-01-04
Stefan Geschke , Menachem Kojman

Borel extensions of Baire measures in ZFC

2011-01-01
Menachem Kojman , Henryk Michalewski
We prove: 1) Every Baire measure on the Kojman&#8211;Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's … We prove: 1) Every Baire measure on the Kojman&#8211;Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Con
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On symmetric indivisbility of countable structures

2011-01-01
Assaf Hasson , Menachem Kojman , Alf Onshuus

Order-theoretic properties of bases in topological spaces I

2010-12-17
Menachem Kojman , David Milovich , Santi Spadaro
The cardinal invariant of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as … The cardinal invariant of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_\omega$ with the countable box topology, $(2^{\aleph_\omega})_\delta$, is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of $\aleph_\omega$. We discuss the influence of principles like $\square_{\aleph_\omega}$ and Chang's conjecture for $\aleph_\omega$ on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms). Within PCF theory we establish the existence of an $(\aleph_4,\aleph_1)$-sparse covering family of countable subsets of $\aleph_\omega$. From this follows an absolute upper bound of $\aleph_4$ on the Noetherian type of $(2^{\aleph_\omega})_\delta$. The proof uses ideas from Shelah's proof that if $\kappa^+ <\lambda$ then his ideal $I[\lambda]$ contains a stationary set consisting of points of cofinality $\kappa$.

Noetherian type in topological products

2010-12-17
Menachem Kojman , David Milovich , Santi Spadaro
The cardinal invariant of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as … The cardinal invariant of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_\omega$ with the countable box topology, $(2^{\aleph_\omega})_\delta$, is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of $\aleph_\omega$. We discuss the influence of principles like $\square_{\aleph_\omega}$ and Chang's conjecture for $\aleph_\omega$ on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms). Within PCF theory we establish the existence of an $(\aleph_4,\aleph_1)$-sparse covering family of countable subsets of $\aleph_\omega$. From this follows an absolute upper bound of $\aleph_4$ on the Noetherian type of $(2^{\aleph_\omega})_\delta$. The proof uses ideas from Shelah's proof that if $\kappa^+ <\lambda$ then his ideal $I[\lambda]$ contains a stationary set consisting of points of cofinality $\kappa$.

Noetherian type in topological products

2010-01-01
Menachem Kojman , David Milovich , Santi Spadaro
The cardinal invariant "Noetherian type" of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as … The cardinal invariant "Noetherian type" of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_\omega$ with the countable box topology, $(2^{\aleph_\omega})_\delta$, is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of $\aleph_\omega$. We discuss the influence of principles like $\square_{\aleph_\omega}$ and Chang's conjecture for $\aleph_\omega$ on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms). Within PCF theory we establish the existence of an $(\aleph_4,\aleph_1)$-sparse covering family of countable subsets of $\aleph_\omega$. From this follows an absolute upper bound of $\aleph_4$ on the Noetherian type of $(2^{\aleph_\omega})_\delta$. The proof uses ideas from Shelah's proof that if $\kappa^+ <\lambda$ then his ideal $I[\lambda]$ contains a stationary set consisting of points of cofinality $\kappa$.
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A symmetrized metric ramsey theorem

2009-09-01
Menachem Kojman

Sharp thresholds for the phase transition between primitive recursive and Ackermannian Ramsey numbers

2008-03-11
Menachem Kojman , Gyesik Lee , Eran Omri +1 more

Metric Baumgartner theorems and universality

2007-01-01
Stefan Geschke , Menachem Kojman
It is consistent with the axioms of set theory that for every metric space X which is isometric to some separable Banach space or to Urysohn's universal separable metric space … It is consistent with the axioms of set theory that for every metric space X which is isometric to some separable Banach space or to Urysohn's universal separable metric space U the following holds:( ) X There exists a nowhere meager subspace of X of cardinality ℵ 1 and any two nowhere meager subsets of X of cardinality ℵ 1 are almost isometric to each other.As a corollary, it is consistent that the Continuum Hypothesis fails and the following hold:(1) There exists an almost-isometry ultrahomogeneous and universal element in the class of separable metric spaces of size ℵ 1 .(2) For every separable Banach space X there exists an almost-isometry conditionally ultrahomogeneous and universal element in the class of subspaces of X of size ℵ 1 .(3) For every finite dimensional Banach space X, there is a unique universal element up to almost-isometry in the class of subspaces of X of size ℵ 1
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Almost isometric embedding between metric spaces

2006-12-01
Menachem Kojman , Saharon Shelah

PCF Theory

2005-01-19
Menachem Kojman

The Threshold for Ackermannian Ramsey numbers

2005-01-01
Menachem Kojman , Eran Omri
For a function $g:\N\to \N$, the \emph{$g$-regressive Ramsey number} of $k$ is the least $N$ so that \[N\stackrel \min \longrightarrow (k)_g\] . This symbol means: for every $c:[N]^2\to \N$ that … For a function $g:\N\to \N$, the \emph{$g$-regressive Ramsey number} of $k$ is the least $N$ so that \[N\stackrel \min \longrightarrow (k)_g\] . This symbol means: for every $c:[N]^2\to \N$ that satisfies $c(m,n)\le g(\min\{m,n\})$ there is a \emph{min-homogeneous} $H\su N$ of size $k$, that is, the color $c(m,n)$ of a pair $\{m,n\}\su H$ depends only on $\min\{m,n\}$. It is known (\cite{km,ks}) that $\id$-regressive Ramsey numbers grow in $k$ as fast as $\Ack(k)$, Ackermann's function in $k$. On the other hand, for constant $g$, the $g$-regressive Ramsey numbers grow exponentially in $k$, and are therefore primitive recursive in $k$. We compute below the threshold in which $g$-regressive Ramsey numbers cease to be primitive recursive and become Ackermannian, by proving: Suppose $g:\N\to \N$ is weakly increasing. Then the $g$-regressive Ramsey numbers are primitive recursive if an only if for every $t&gt;0$ there is some $M_t$ so that for all $n\ge M_t$ it holds that $g(m)

Small sets in convex geometry and formal independence over ZFC

2005-01-01
Menachem Kojman
To each closed subset S of a finite‐dimensional Euclidean space corresponds a σ ‐ideal of sets 𝒥 ( S ) which is σ ‐generated over S by the convex subsets … To each closed subset S of a finite‐dimensional Euclidean space corresponds a σ ‐ideal of sets 𝒥 ( S ) which is σ ‐generated over S by the convex subsets of S . The set‐theoretic properties of this ideal hold geometric information about the set. We discuss the relation of reducibility between convexity ideals and the connections between convexity ideals and other types of ideals, such as the ideals which are generated over squares of Polish space by graphs and inverses of graphs of continuous self‐maps, or Ramsey ideals, which are generated over Polish spaces by the homogeneous sets with respect to some continuous pair coloring. We also attempt to present to nonspecialists the set‐theoretic methods for dealing with formal independence as a means of geometric investigations.
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PCF Theory

2005-01-01
Menachem Kojman
This article is a very short introduction to pcf theory for topologists. This article is a very short introduction to pcf theory for topologists.

CONTINUOUS RAMSEY THEORY ON POLISH SPACES AND COVERING THE PLANE BY FUNCTIONS

2004-12-01
Stefan Geschke , Martin Goldstern , Menachem Kojman
We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: … We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: see text] of a pair-coloring c:[X] 2 →2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2 ω , c min and c max , which satisfy [Formula: see text] and prove: Theorem. (1) For every Polish space X and every continuous pair-coloringc:[X] 2 →2with[Formula: see text], [Formula: see text] (2) There is a model of set theory in which[Formula: see text]and[Formula: see text]. The consistency of [Formula: see text] and of [Formula: see text] follows from [20]. We prove that [Formula: see text] is equal to the covering number of (2 ω ) 2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c min gives: Theorem. There is a model of set theory in which (1) ℝ 2 is coverable byℵ 1 graphs and reflections of graphs of continuous real functions; (2) ℝ 2 is not coverable byℵ 1 graphs and reflections of graphs of Lipschitz real functions. Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes [Formula: see text] from row (3).
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On two problems of Erdos and Hechler: New methods in singular madness

2004-06-21
Menachem Kojman , Wiesław Kubiś , Saharon Shelah
For an infinite cardinal $\mu$, $\operatorname {MAD}(\mu )$ denotes the set of all cardinalities of nontrivial maximal almost disjoint families over $\mu$. Erdős and Hechler proved in 1973 the consistency … For an infinite cardinal $\mu$, $\operatorname {MAD}(\mu )$ denotes the set of all cardinalities of nontrivial maximal almost disjoint families over $\mu$. Erdős and Hechler proved in 1973 the consistency of $\mu \in \operatorname {MAD}(\mu )$ for a singular cardinal $\mu$ and asked if it was ever possible for a singular $\mu$ that $\mu \notin \operatorname {MAD}(\mu )$, and also whether $2^{\operatorname {cf}\mu } <\mu \Longrightarrow \mu \in \operatorname {MAD}(\mu )$ for every singular cardinal $\mu$. We introduce a new method for controlling $\operatorname {MAD} (\mu )$ for a singular $\mu$ and, among other new results about the structure of $\operatorname {MAD}(\mu )$ for singular $\mu$, settle both problems affirmatively.
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Almost isometric embeddings of metric spaces

2004-01-01
Menachem Kojman , Saharon Shelah
We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. … We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum separable metric spaces on aleph_1 so that every separable metric space is almost isometrically embedded into one of them when the continuum hypothesis fails. (3) There is no collection of fewer than continuum metric spaces of cardinality aleph_2 so that every ultra-metric space of cardinality aleph_2 is almost isometrically embedded into one of them if aleph_2&lt;2^{aleph_0}. We also prove that various spaces X satisfy that if a space X is almost isometric to X than Y is isometric to X.

Sequentially linearly Lindelöf spaces

2003-02-01
Menachem Kojman , Victoria Lubitch

Van der Waerden spaces and Hindman spaces are not the same

2002-12-16
Menachem Kojman , Saharon Shelah
A Hausdorff topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>van der Waerden</italic> if for every sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis … A Hausdorff topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>van der Waerden</italic> if for every sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n element-of omega"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_{n\in \omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there is a converging subsequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n element-of upper A"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_{n\in A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A subset-of-or-equal-to omega"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A\subseteq \omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains arithmetic progressions of all finite lengths. A Hausdorff topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>Hindman</italic> if for every sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n element-of omega"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_{n\in \omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there is an <italic>IP-converging</italic> subsequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n element-of upper F upper S left-parenthesis upper B right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>F</mml:mi> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_{n\in FS(B)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some infinite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B subset-of-or-equal-to omega"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">B\subseteq \omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.

Convex decompositions in the plane and continuous pair colorings of the irrationals

2002-12-01
Stefan Geschke , Menachem Kojman , Wiesław Kubiś +1 more

Thomas Jech. <i>Singular cardinal problem: Shelah's theorem on 2<sup>ℵ<sub>ω</sub></sup></i>. Bulletin of the London Mathematical Society, vol. 24 (1992), pp. 127–139.

2002-06-01
Menachem Kojman
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Continuous Ramsey theory on Polish spaces and covering the plane by functions

2002-05-31
Stefan Geschke , Martin Goldstern , Menachem Kojman
We investigate the Ramsey theory of continuous pair-colorings on complete, separable metric spaces, and apply the results to the problem of covering a plane by functions. The homogeneity number hm(c) … We investigate the Ramsey theory of continuous pair-colorings on complete, separable metric spaces, and apply the results to the problem of covering a plane by functions. The homogeneity number hm(c) of a pair-coloring c:[X]^2 -> 2 is the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2^omega, c_min and c_max, which satisfy hm(c_min)\le hm(c_max) and prove: 1. For every Polish space X and every continuous pair-coloring c:[X]^2 -> 2 with hm(c) uncountable: hm(c)= hm(c_min) or hm(c)=hm(c_max) 2. There is a model of set theory in which hm(c_min)=aleph_1 and hm(c_max)=aleph_2 (The consistency of hm(c_min) = 2^aleph0 and of hm(c_max) < 2^aleph0 is known) We prove that hm(c_min) is equal to the covering number of (2^omega)^2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c_min gives: There is a model of set theory in which 1. R^2 is coverable by aleph1 graphs and reflections of graphs of continuous real functions; 2. R^2 is not coverable by aleph1 graphs and reflections of graphs of Lipschitz real functions.

Continuous Ramsey theory on Polish spaces and covering the plane by functions

2002-01-01
Stefan Geschke , Martin Goldstern , Menachem Kojman
We investigate the Ramsey theory of continuous pair-colorings on complete, separable metric spaces, and apply the results to the problem of covering a plane by functions. The homogeneity number hm(c) … We investigate the Ramsey theory of continuous pair-colorings on complete, separable metric spaces, and apply the results to the problem of covering a plane by functions. The homogeneity number hm(c) of a pair-coloring c:[X]^2 -> 2 is the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2^omega, c_min and c_max, which satisfy hm(c_min)\le hm(c_max) and prove: 1. For every Polish space X and every continuous pair-coloring c:[X]^2 -> 2 with hm(c) uncountable: hm(c)= hm(c_min) or hm(c)=hm(c_max) 2. There is a model of set theory in which hm(c_min)=aleph_1 and hm(c_max)=aleph_2 (The consistency of hm(c_min) = 2^aleph0 and of hm(c_max) < 2^aleph0 is known) We prove that hm(c_min) is equal to the covering number of (2^omega)^2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c_min gives: There is a model of set theory in which 1. R^2 is coverable by aleph1 graphs and reflections of graphs of continuous real functions; 2. R^2 is not coverable by aleph1 graphs and reflections of graphs of Lipschitz real functions.
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Hindman spaces

2001-12-20
Menachem Kojman
A topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>Hindman</italic> if for every sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n … A topological space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>Hindman</italic> if for every sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists an infinite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D subset-of-or-equal-to double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊆</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">D\subseteq \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that the sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x Subscript n Baseline right-parenthesis Subscript n element-of upper F upper S left-parenthesis upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>F</mml:mi> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(x_n)_{n\in FS(D)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, indexed by all finite sums over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is IP-converging in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Not all sequentially compact spaces are Hindman. The product of two Hindman spaces is Hindman. Furstenberg and Weiss proved that all compact metric spaces are Hindman. We show that every Hausdorff space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that satisfies the following condition is Hindman: <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis reverse-solidus quad The closure of every countable set in upper X is compact and first hyphen countable period reverse-solidus quad"> <mml:semantics> <mml:mrow> <mml:mtext>(</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mtext>)\quad The closure of every countable set in </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>X</mml:mi> </mml:mrow> <mml:mtext> is compact and first-countable.\quad </mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\text {($*$)\quad The closure of every countable set in $X$ is compact and first-countable.\quad }</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> Consequently, there exist nonmetrizable and noncompact Hindman spaces. The following is a particular consequence of the main result: every bounded sequence of monotone (not necessarily continuous) real functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has an IP-converging subsequences.

Cantor-Bendixson degrees and convexity in ℝ2

2001-12-01
Menachem Kojman

Van der Waerden spaces

2001-08-28
Menachem Kojman
A topological space $X$ is van der Waerden if for every sequence $(x_n)_n$ in $X$ there exists a converging subsequence $(x_{n_k})_k$ so that $\{{n_k}:k\in \mathbb {N}\}$ contains arbitrarily long finite … A topological space $X$ is van der Waerden if for every sequence $(x_n)_n$ in $X$ there exists a converging subsequence $(x_{n_k})_k$ so that $\{{n_k}:k\in \mathbb {N}\}$ contains arbitrarily long finite arithmetic progressions. Not every sequentially compact space is van der Waerden. The product of two van der Waerden spaces is van der Waerden. The following condition on a Hausdorff space $X$ is sufficent for $X$ to be van der Waerden: [$(*)$] The closure of every countable set in $X$ is compact and first-countable. A Hausdorff space $X$ that satisfies $(*)$ satisfies, in fact, a stronger property: for every sequence $(x_n)$ in $X$: [$(\star )$] There exists $A\subseteq \mathbb {N}$ so that $(x_n)_{n\in A}$ is converging, and $A$ contains arbitrarily long finite arithmetic progressions and sets of the form $FS(D)$ for arbitrarily large finite sets $D$. There are nonmetrizable and noncompact spaces which satisfy $(*)$. In particular, every sequence of ordinal numbers and every bounded sequence of real monotone functions on $[0,1]$ satisfy $(\star )$.
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Fallen cardinals

2001-05-01
Menachem Kojman , Saharon Shelah

Countably convex G<sub>δ</sub>sets

2001-01-01
V. P. Fonf , Menachem Kojman

Van der Waerden spaces and Hindman spaces are not the same

2001-01-01
Menachem Kojman , Saharon Shelah
A Hausdorff topological space X is van der Waerden if for every sequence (x_n)_n in X there is a converging subsequence (x_n)_{n in A} where subset A of omega contains … A Hausdorff topological space X is van der Waerden if for every sequence (x_n)_n in X there is a converging subsequence (x_n)_{n in A} where subset A of omega contains arithmetic progressions of all finite lengths. A Hausdorff topological space X is Hindman if for every sequence (x_n)_n in X there is an IP-converging subsequence (x_n)_{n in FS(B)} for some infinite subset B of omega. We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.

Fallen Cardinals

2000-09-07
Menachem Kojman , Saharon Shelah
We prove that for every singular cardinal mu of cofinality omega, the complete Boolean algebra compP_mu(mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra Comp Col(omega_1,mu^{aleph_0}). … We prove that for every singular cardinal mu of cofinality omega, the complete Boolean algebra compP_mu(mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra Comp Col(omega_1,mu^{aleph_0}). Consequently, adding a generic filter to the quotient algebra P_mu(mu)=P(mu)/[mu]^{<mu} collapses mu^{aleph_0} to aleph_1. Another corollary is that the Baire number of the space U(mu) of all uniform ultrafilters over mu is equal to omega_2. The corollaries affirm two conjectures by Balcar and Simon. The proof uses pcf theory.

Convexity ranks in higher dimensions

2000-01-01
Menachem Kojman
A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed … A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure
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Coauthor Papers Together
Saharon Shelah 33
Martin Goldstern 11
Juris Steprāns 9
Assaf Rinot 8
Stefan Geschke 7
Santi Spadaro 4
David Milovich 4
Rami Grossberg 2
Eran Omri 2
Wiesław Kubiś 2
Alf Onshuus 1
Micha A. Perles 1
William Chen-Mertens 1
Victoria Lubitch 1
Henryk Michalewski 1
Assaf Hasson 1
Andreas Weiermann 1
R. Grossberg 1
René Schipperus 1
V. P. Fonf 1
Gyesik Lee 1
Stefanie Huber 1

Nonexistence of universal orders in many cardinals

1992-09-01
Menachem Kojman , Saharon Shelah
Abstract Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which … Abstract Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ 1 ; without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove—again in ZFC—that for a large class of cardinals there is no universal linear order (e.g. in every regular ). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles” ℵ 1 —a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p -adic rings and fields, partial orders, models of PA and so on).
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A normal space X for which X×I is not normal

1971-01-01
Marry Rudin
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Shelah's pcf theory and its applications

1990-12-01
Maxim R. Burke , Menachem Magidor
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Exact upper bounds and their uses in set theory

1998-08-01
Menachem Kojman

Partitioning pairs of countable ordinals

1987-01-01
Stevo Todorčević
We show that the pairs of countable ordinals can be colored with uncountably many s colors so that every uncountable set contains pairs of every color.This gives a definitive limitation … We show that the pairs of countable ordinals can be colored with uncountably many s colors so that every uncountable set contains pairs of every color.This gives a definitive limitation on any form of a Ramsey Theorem for the uncountable which reduces the set of colors on some uncountable square.The first such limitation was given by Sierpifiski [21] for only two colors.This was later improved by Laver (see [13]) to three colors and then by Galvin and Shelah [4] to four colors (see also Blass [1]).Our method is not based on the existence of certain uncountable linear orderings (an approach still of interest) as was the case with [21], [13], [4] and [1], but on a fine analysis of the concept of a special Aronszajn tree.This analysis will give us also a new
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CHAIN CONDITIONS OF PRODUCTS, AND WEAKLY COMPACT CARDINALS

2014-09-01
Assaf Rinot
Abstract The history of productivity of the κ -chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly … Abstract The history of productivity of the κ -chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal $\kappa &gt; \aleph _1 {\rm{,}}$ the principle □( k ) is equivalent to the existence of a certain strong coloring $c\,:\,[k]^2 \, \to $ k for which the family of fibers ${\cal T}\left( c \right)$ is a nonspecial κ -Aronszajn tree. The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if the κ -chain condition is productive for a given regular cardinal $\kappa &gt; \aleph _1 {\rm{,}}$ then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ -chain condition is productive.

The Singular Cardinal Hypothesis Revisited

1992-01-01
Moti Gitik , Menachem Magidor

The universality spectrum of stable unsuperstable theories

1992-07-01
Menachem Kojman , Saharon Shelah
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Partition relations for cardinal numbers

1965-03-01
Péter L. Erdős , A. Hajnal , R. Rado
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Powers of regular cardinals

1970-01-01
William B. Easton

Advances in Cardinal Arithmetic

1993-01-01
Saharon Shelah

A ZFC Dowker space in ℵ_{𝜔+1}: An application of pcf theory to topology

1998-01-01
Menachem Kojman , Saharon Shelah
The existence of a Dowker space of cardinality<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript omega plus 1"><mml:semantics><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ω</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\aleph _{\omega +1}</mml:annotation></mml:semantics></mml:math></inline-formula>and weight<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript omega plus 1"><mml:semantics><mml:msub><mml:mi … The existence of a Dowker space of cardinality<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript omega plus 1"><mml:semantics><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ω</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\aleph _{\omega +1}</mml:annotation></mml:semantics></mml:math></inline-formula>and weight<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript omega plus 1"><mml:semantics><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ω</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:annotation encoding="application/x-tex">\aleph _{\omega +1}</mml:annotation></mml:semantics></mml:math></inline-formula>is proved in ZFC using pcf theory.
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Universal structures in power ℵ<sub>1</sub>

1990-06-01
Alan H. Mekler
Abstract It is consistent with ¬ CH that every universal theory of relational structures with the joint embedding property and amalgamation for - (3)-diagrams has a universal model of cardinality … Abstract It is consistent with ¬ CH that every universal theory of relational structures with the joint embedding property and amalgamation for - (3)-diagrams has a universal model of cardinality ℵ 1 . For classes with amalgamation for - (4)-diagrams it is consistent that and there is a universal model of cardinality ℵ 2 .

On universal graphs without instances of CH

1984-02-01
Saharon Shelah

Splitting strongly almost disjoint families

1986-01-01
A. Hajnal , István Juhász , Saharon Shelah
We say that a family $\mathcal {A} \subset {[\lambda ]^\kappa }$ is strongly almost disjoint if something more than just $|A \cap B| < \kappa$, e.g. that $|A \cap B| … We say that a family $\mathcal {A} \subset {[\lambda ]^\kappa }$ is strongly almost disjoint if something more than just $|A \cap B| < \kappa$, e.g. that $|A \cap B| < \sigma < \kappa$, is assumed for $A$, $B \in \mathcal {A}$. We formulate conditions under which every such strongly a.d. family is "essentially disjoint", i.e. for each $A \in \mathcal {A}$ there is $F(A) \in {[A]^{ < \kappa }}$ so that $\{ A\backslash F(A):A \in \mathcal {A}\}$ is disjoint. On the other hand, we get from a supercompact cardinal the consistency of ${\text {GCH}}$ plus the existence of a family $\mathcal {A} \subset {[{\omega _{\omega + 1}}]^{{\omega _1}}}$ whose elements have pairwise finite intersections and such that it does not even have property $B$. This solves an old problem raised in [
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The generalized continuum hypothesis revisited

2000-12-01
Saharon Shelah
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Αll $ℵ_1$-dense sets of reals can be isomorphic

1973-01-01
James E. Baumgartner
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Many simple cardinal invariants

1993-05-01
Martin Goldstern , Saharon Shelah
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Finite sums from sequences within cells of a partition of N

1974-07-01
Neil Hindman

Comparing almost-disjoint families

1986-09-01
Péter Komjáth

On a property of families of sets

1964-03-01
Péter L. Erdős , A. Hajnal

Sets in a euclidean space which are not a countable union of convex subsets

1990-10-01
Menachem Kojman , Micha A. Perles , Saharon Shelah

Convexity ranks in higher dimensions

2000-01-01
Menachem Kojman
A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed … A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure
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NO BOUND FOR THE FIRST FIXED POINT

2005-12-01
Moti Gitik
Our aim is to show that it is impossible to find a bound for the power of the first fixed point of the aleph function. Our aim is to show that it is impossible to find a bound for the power of the first fixed point of the aleph function.
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Shelah’s revised GCH theorem and a question by Alon on infinite graphs colorings

2014-02-07
Menachem Kojman

Knaster and friends II: The C-sequence number

2020-06-20
Chris Lambie‐Hanson , Assaf Rinot
Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the [Formula: see text]-sequence number, which can be seen as a measure of … Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the [Formula: see text]-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of [Formula: see text] and independence results about the [Formula: see text]-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general [Formula: see text]-sequence spectrum and uncover some tight connections between the [Formula: see text]-sequence spectrum and the strong coloring principle [Formula: see text], introduced in Part I of this series.
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A Lindelöf group with non-Lindelöf square

2017-12-06
Yinhe Peng , Liuzhen Wu

When does almost free imply free? (For groups, transversals, etc.)

1994-01-01
Menachem Magidor , Saharon Shelah
We show that the construction of an almost free nonfree Abelian group can be pushed from a regular cardinal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ<!-- κ --></mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> … We show that the construction of an almost free nonfree Abelian group can be pushed from a regular cardinal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ<!-- κ --></mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript kappa plus 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>κ<!-- κ --></mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _{\kappa + 1}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Hence there are unboundedly many almost free nonfree Abelian groups below the first cardinal fixed point. We give a sufficient condition for “ <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ<!-- κ --></mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> free implies free”, and then we show, assuming the consistency of infinitely many supercompacts, that one can have a model of ZFC+G.C.H. in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript omega squared plus 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _{{\omega ^2} + 1}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> free implies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript omega squared plus 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _{{\omega ^2} + 2}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> free. Similar construction yields a model in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef Subscript kappa"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mi>κ<!-- κ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _\kappa }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> free implies free for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ<!-- κ --></mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the first cardinal fixed point (namely, the first cardinal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α<!-- α --></mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha equals normal alef Subscript alpha"> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mi>α<!-- α --></mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha = {\aleph _\alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). The absolute results about the existence of almost free nonfree groups require only minimal knowledge of set theory. Also, no knowledge of metamathematics is required for reading the section on the combinatorial principle used to show that almost free implies free. The consistency of the combinatorial principle requires acquaintance with forcing techniques.
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Chang’s conjecture for ℵω

1990-06-01
Jean-Pierre Levinski , Menachem Magidor , Saharon Shelah

Universal graphs without instances of CH: Revisited

1990-02-01
Saharon Shelah

Families close to disjoint ones

1984-09-01
Péter Komjáth

Almost-disjoint sets the dense set problem and the partition calculus

1976-05-01
James E. Baumgartner

Extending partial isomorphisms of graphs

1992-12-01
Ehud Hrushovski

THE SINGULAR CARDINALS PROBLEM; INDEPENDENCE RESULTS

1983-10-13
Saharon Shelah
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Notes on Singular Cardinal Combinatorics

2005-07-01
James Cummings
We present a survey of combinatorial set theory relevant to the study of singular cardinals and their successors. The topics covered include diamonds, squares, club guessing, forcing axioms, and PCF … We present a survey of combinatorial set theory relevant to the study of singular cardinals and their successors. The topics covered include diamonds, squares, club guessing, forcing axioms, and PCF theory.
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A note on Noetherian type of spaces

2010-01-01
Lajos Soukup
The Noetherian type of a space X, Nt(X), is the least cardinal kappa such that X has a base B such that every element of the base is contained in … The Noetherian type of a space X, Nt(X), is the least cardinal kappa such that X has a base B such that every element of the base is contained in less than kappa many elements of the base. Denote X the space obtained from 2^{aleph_omega} by declaring the G_delta sets to be open. Milovich proved that if Square_{aleph_omega} holds and (aleph_omega)^omega=aleph_{omega+1} then Nt(X)=omega_1. Answering a question of Spadaro, we show that if (aleph_omega)^omega=aleph_{omega+1} and a strong form of Chang Conjecture holds for aleph_ωthen Nt(X)&gt;omega_1.

A solution to the L space problem

2005-12-21
Justin Tatch Moore
In this paper I will construct a non-separable hereditarily Lindelöf space (L space) without any additional axiomatic assumptions. The constructed space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow … In this paper I will construct a non-separable hereditarily Lindelöf space (L space) without any additional axiomatic assumptions. The constructed space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a subspace of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T Superscript omega 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathbb {T}}^{\omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the unit circle. It is shown to have a number of properties which may be of additional interest. For instance it is shown that the closure in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T Superscript omega 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {T}^{\omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of any uncountable subset of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a canonical copy of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T Superscript omega 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {T}^{\omega _1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. I will also show that there is a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon left-bracket omega 1 right-bracket squared right-arrow omega 1"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">f:[\omega _1]^2 \to \omega _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A comma upper B subset-of-or-equal-to omega 1"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">A,B \subseteq \omega _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are uncountable and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi greater-than omega 1"> <mml:semantics> <mml:mrow> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\xi &gt; \omega _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then there are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha greater-than beta"> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi>β<!-- β --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha &gt; \beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> respectively with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis alpha comma beta right-parenthesis equals xi"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>ξ<!-- ξ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f (\alpha ,\beta ) = \xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Previously it was unknown whether such a function existed even if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega 1"> <mml:semantics> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> was replaced by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">ℵ<!-- ℵ --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\aleph _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The results all stem from the analysis of oscillations of coherent sequences <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle e Subscript beta Baseline colon beta greater-than omega 1 mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨<!-- ⟨ --></mml:mo> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>β<!-- β --></mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo fence="false" stretchy="false">⟩<!-- ⟩ --></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle e_\beta :\beta &gt; \omega _1\rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of finite-to-one functions. I expect that the methods presented will have other applications as well.
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Ordered spaces with special bases

1998-01-01
Harold Bennett , David Lutzer
We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered … We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space h
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Embedding theorems for graphs establishing negative partition relations

1978-09-01
P. Erdős , A. Hajnal
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A Mathematical Incompleteness in Peano Arithmetic

1977-01-01
J. B. Paris

The Monadic Theory of Order

1975-11-01
Saharon Shelah
We deal with the monadic (second-order) theory of order.We prove all known results in a unified way, show a general way of reduction, prove more results and show the limitation … We deal with the monadic (second-order) theory of order.We prove all known results in a unified way, show a general way of reduction, prove more results and show the limitation on extending them.We prove (CH) that the monadic theory of the real order is undecidable.Our methods are model-theoretic, and we do not use automaton theory.This is a slightly corrected version of a very old work.
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C-scattered and paracompact spaces

1971-01-01
Rastislav Telgársky
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Cardinal arithmetic for skeptics

1992-04-01
Saharon Shelah
When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense.The resulting preoccupation with "consistency" … When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense.The resulting preoccupation with "consistency" rather than "truth" may be felt to give the subject an air of unreality.Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of 2H°, cannot be settled on the basis of the usual axioms of set theory (ZFC).Although much can be said in favor of such independence results, rather than undertaking to challenge such prejudices, we have a more modest goal; we wish to point out an area of contemporary set theory in which theorems are abundant, although the conventional wisdom views the subject as dominated by independence results, namely, cardinal arithmetic.To see the subject in this light it will be necessary to carry out a substantial shift in our point of view.To make a very rough analogy with another generalization of ordinary arithmetic, the natural response to the loss of unique factorization caused by moving from Z to other rings of algebraic integers is to compensate by changing the definitions, rescuing the theorems.Similarly, after shifting the emphasis in cardinal arithmetic from the usual notion of exponentiation to a somewhat more subtle variant, a substantial body of results is uncovered that leads to new theorems in cardinal arithmetic and has applications in other areas as well.The first shift is from cardinal exponentiation to the more general notion of an infinite product of infinite cardinals; the second shift is from cardinality to cofinality; and the final shift is from true cofinality to potential cofinality (pcf).The first shift is quite minor and will be explained in §1.The main shift in viewpoint will be presented in §4 after a review of basics in §1, a brief look at history in §2, and some personal history in §3.The main results on pcf are presented in §5.Applications to cardinal arithmetic are described in §6.The limitations on independence proofs are discussed in §7, and in §8 we discuss the status of two axioms that arise in the new setting.Applications to other areas are found in §9.The following result is a typical application of the theory.Theorem A. // 2N" < Xw for all n then 2N°-< N^ .The subscript 4 occurring here is admittedly very strange.Our thesis is that the theorem cannot really be understood in the framework of conventional cardinal arithmetic, but that it makes excellent sense as a theorem on pcf.Another way of putting the matter is that the theory of cardinal arithmetic involves two quite different aspects, one of which is totally independent of the usual axioms
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On the Singular Cardinals Problem

2003-08-01
Jack H. Silver

Homogeneous Families and their Automorphism Groups

1995-10-01
Menachem Kojman , Saharon Shelah
A homogeneous family of subsets over a given set is one with a very ‘rich’ automorphism group. We prove the existence of bi-universal element in the class of homogeneous families … A homogeneous family of subsets over a given set is one with a very ‘rich’ automorphism group. We prove the existence of bi-universal element in the class of homogeneous families over a given infinite set and give an explicit construction of 2 2 ℵ 0 isomorphism types of homogeneous families over a countable set.
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Collapsing successors of singulars

1997-01-01
James Cummings
Let $\kappa$ be a singular cardinal in $V$, and let $W \supseteq V$ be a model such that $\kappa ^+_V = \lambda ^+_W$ for some $W$-cardinal $\lambda$ with $W \models … Let $\kappa$ be a singular cardinal in $V$, and let $W \supseteq V$ be a model such that $\kappa ^+_V = \lambda ^+_W$ for some $W$-cardinal $\lambda$ with $W \models \operatorname {cf}(\kappa ) \neq \operatorname {cf}(\lambda )$. We apply Shelah's pcf theory to study this situation, and prove the following results. $W$ is not a $\kappa ^+$-c.c generic extension of $V$. There is no "good scale for $\kappa$" in $V$, so in particular weak forms of square must fail at $\kappa$. If $V \models \operatorname {cf}(\kappa ) = \aleph _0$ then $V \models {}$ "$\kappa$ is strong limit $\implies 2^\kappa = \kappa ^+$", and also ${}^\omega \kappa \cap W \supsetneq {}^\omega \kappa \cap V$. If $\kappa = \aleph _\omega ^V$ then $\lambda \le (2^{\aleph _0})_V$.
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Independence of strong partition relation for small cardinals, and the free-subset problem

1980-09-01
Saharon Shelah
Abstract We prove the independence of a strong partition relation on ℵ ω , answering a question of Erdös and Hajnal. We then give an almost complete answer to the … Abstract We prove the independence of a strong partition relation on ℵ ω , answering a question of Erdös and Hajnal. We then give an almost complete answer to the free subset problem.

Singular Cardinal Problem: Shelah's Theorem on 2ℵω

1992-03-01
Thomas Jech
This is an expository paper giving a complete proof of a theorem of Saharon Shelah: if 2 ℵ n < ℵ ω for all n < ω, then 2 ℵ … This is an expository paper giving a complete proof of a theorem of Saharon Shelah: if 2 ℵ n < ℵ ω for all n < ω, then 2 ℵ ω < ℵ ω 4 .

A very weak square principle

1997-03-01
Matthew Foreman , Menachem Magidor
In this paper we explicate a very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L . This principle is strong … In this paper we explicate a very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L . This principle is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals. In this section we show that this principle is equivalent to a common combinatorial device, which we call a Jensen matrix. In the second section we show that our principle is consistent with a supercompact cardinal. In the third section of this paper we show that this principle is exactly equivalent to the statement that every torsion free Abelian group has a filtration into σ -balanced subgroups. In the fourth section of this paper we show that this principle fails if you assume the Chang's Conjecture: In the fifth section of the paper we review the proofs that the various weak squares we consider are strictly decreasing in strength. Section 6 was added in an ad hoc manner after the rest of the paper was written, because the subject matter of Theorem 6.1 fit well with the rest of the paper. It deals with a principle dubbed “Not So Very Weak Square”, which appears close to Very Weak Square but turns out not to be equivalent.

A condition for a compact plane set to be a union of finitely many convex sets

1974-07-01
H. G. Eggleston
All the sets with which we are concerned are subsets of the real Euclidean plane E 2 . By L m we denote those subsets X of E 2 for … All the sets with which we are concerned are subsets of the real Euclidean plane E 2 . By L m we denote those subsets X of E 2 for which, if p l , p 2 , …, P m are any m points of X , then at least one segment p i p j , i ≠ j consists entirely of points of X . L 2 is the class of convex subsets of E 2 . We shall show that if X is closed and X ∈ L m . then X is the union of finitely many convex sets. This extends a result of Valentine (4). See also (1),(2),(3).