On Product Schur Triples in the Integers

This paper investigates Ramsey-type problems for “product Schur triples” – ordered triples of positive integers (a,b,c) such that ab=c – in analogy to the well-studied (additive) Schur triples where a+b=c.

Significance:
The work significantly extends classical Ramsey theory on integers, which traditionally focuses on linear equations, to a multiplicative setting. Schur’s theorem (1917) guarantees a monochromatic solution to a+b=c in any finite coloring of [n] (for n large enough). This paper pioneers a systematic study of the corresponding questions for ab=c:
1. How large must a subset of [2,n] be to guarantee a monochromatic product Schur triple under any k-coloring?
2. What is the minimum number of monochromatic product Schur triples in any k-coloring of [2,n]?
3. What is the probability threshold for a random subset of [2,n] to contain a product Schur triple?
4. How densely must a pre-existing set be perturbed with random elements to ensure a monochromatic product Schur triple?
The findings are particularly noteworthy as they provide some of the first results for nonlinear equations in random and randomly-perturbed settings, opening up new avenues in combinatorial number theory.

Key Innovations and Main Results:

1. Deterministic Setting - Size of Product Schur-Ramsey Sets:

   • The paper defines g*(k,n) as the minimum size s such that any subset A of [2,n] of size s must contain a monochromatic product Schur triple under any k-coloring.
   • Innovation: It links g*(k,n) to both the classical Schur number S(k) and the “double-sum Schur number” S'(k) (the minimum m such that any k-coloring of [m] has a monochromatic solution to a+b=c or a+b=c-1, previously studied by Abbott and Hanson).
   • Result (Theorem 1.1): Provides bounds: n - n^(1/S'(k)) <= g*(k,n) <= n - (1-epsilon)n^(1/S(k)). These bounds are nearly tight for k=1,2,3 where S'(k)=S(k).

2. Deterministic Setting - Counting Monochromatic Triples:

   • Innovation: Addresses the quantitative question of how many monochromatic product Schur triples must exist.
   • Result (Theorem 1.2): Any 2-coloring of [2,n] contains at least n^(1/3 - epsilon) monochromatic product Schur triples. (The authors note a concurrent, stronger result by Aragão, Chapman, Ortega, and Souza achieving n^(1/2) polylog(n)).

3. Random Setting - Threshold for Existence:

   • Innovation: Determines the threshold probability p for a random subset [2,n]_p (where each element is chosen with probability p) to contain a product Schur triple. This is a novel contribution for nonlinear equations.
   • Result (Theorem 1.3): The threshold is (n log n)^(-1/3). This contrasts with the n^(-2/3) threshold for additive Schur triples.

4. Randomly Perturbed Setting - Threshold for Existence:

   • Innovation: Studies the scenario where a dense deterministic set C_n (of size (1-alpha(n))n) is augmented by a random set [2,n]_p. The goal is to find the threshold p_alpha for C_n U [2,n]_p to contain a product Schur triple.
   • Result (Theorem 1.4 & 1.5):
     • For a specific range of initial densities alpha (decaying slightly faster than (log n)^(-delta), where delta ≈ 0.086 is related to Ford’s constant on the distribution of divisors), the threshold p_alpha(n) is n^(-1/2 + o(1)).
     • More generally, the paper provides bounds for p_alpha(n) that interpolate between the purely random case (roughly n^(-1/3)) and the dense perturbed case (n^(-1/2)), depending on the initial density alpha. The analysis involves functions f(alpha) and beta(alpha) relating alpha to exponents in the threshold.

Main Prior Ingredients Needed:

1. Schur’s Theorem and Schur Numbers (S(k)): The foundational result stating that for any k, [S(k)] is the smallest interval [N] such that any k-coloring of [N] contains a monochromatic solution to a+b=c. The concept and bounds on S(k) are central.
2. Abbott and Wang’s Problem: Their work on g(k,n), the size of the largest subset of [n] that avoids a monochromatic a+b=c under k-coloring, provides the template for defining g*(k,n).
3. Double-Sum Schur Numbers (S’(k)): Introduced by Abbott and Hanson in the context of “strongly sum-free sets,” these numbers naturally appear in the lower bounds for g*(k,n).
4. Probabilistic Method in Combinatorics: Techniques like the first and second moment methods, Chernoff bounds, and Chebyshev’s inequality are used for analyzing random sets.
5. Theory of Threshold Functions for Random Structures: The concept of a threshold probability for a monotone property (Bollobás and Thomason).
6. Results on Divisor Functions: Classical bounds on the number of divisors of an integer (e.g., Wigert’s result d(m) <= m^(O(1/log log m))) are used in counting arguments.
7. Ford’s Work on Integers with Divisors in an Interval: Ford’s results on the count H(n,I) of integers up to n having a divisor in an interval I are crucial for the analysis in the randomly perturbed setting, leading to the appearance of the constant delta.
8. Ramsey Theory in Randomly Perturbed Sets: Prior work by Aigner-Horev and Person on monochromatic additive Schur triples in randomly perturbed dense sets of integers serves as inspiration.