On Rainbow Matchings for Hypergraphs
On Rainbow Matchings for Hypergraphs
For any positive integer $m$, let $[m]:=\{1,\ldots,m\}$. Let $n,k,t$ be positive integers. Aharoni and Howard conjectured that if, for $i\in [t]$, $\mathcal{F}_i\subseteq [n]^k:= \{(a_1,\ldots,a_k): a_j\in [n] \mbox{ for } j\in [k]\}$ and $|\mathcal{F}_i|>(t-1)n^{k-1}$, then there exists $M\subseteq [n]^k$ such that $|M|=t$, $|M\cap \mathcal{F}_i|=1$ for $i\in [t]$ and $A\cap B=\emptyset$ for …