On rainbow matchings for hypergraphs

For any posotive integer $m$, let $[m]:=\{1,\ldots,m\}$. Let $n,k,t$ be positive integers. Aharoni and Howard conjectured that if, for $i\in [t]$, $\mathcal{F}_i\subset[n]^k:= \{(a_1,\ldots,a_k): a_j\in [n] \mbox{ for } j\in [k]\}$ and $|\mathcal{F}_i|>(t-1)n^{k-1}$, then there exist $M\subseteq [n]^k$ such that $|M|=t$ and $|M\cap \mathcal{F}_i|=1$ for $i\in [t]$ We show that this conjecture holds when $n\geq 3(k-1)(t-1)$. Let $n, t, k_1\ge k_2\geq \ldots\geq k_t$ be positive integers. Huang, Loh and Sudakov asked for the maximum $Π_{i=1}^t |{\cal R}_i|$ over all ${\cal R}=\{{\cal R}_1, \ldots ,{\cal R}_t\}$ such that each ${\cal R}_i$ is a collection of $k_i$-subsets of $[n]$ for which there does not exist a collection $M$ of subsets of $[n]$ such that $|M|=t$ and $|M\cap \mathcal{R}_i|=1$ for $i\in [t]$ %and ${\cal R}$ does not admit a rainbow matching. We show that for sufficiently large $n$ with $\sum_{i=1}^t k_i\leq n(1-(4k\ln n/n)^{1/k})$, $\prod_{i=1}^t |\mathcal{R}_i|\leq {n-1\choose k_1-1}{n-1\choose k_2-1}\prod_{i=3}^{t}{n\choose k_i}$. This bound is tight.

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