Extremal Problem for Matchings and Rainbow Matchings on Direct Products

TL;DR

This paper establishes extremal bounds for matchings and rainbow matchings in product hypergraphs, providing conditions under which maximum sizes are achieved and characterizing rainbow matching free families.

Contribution

It introduces new bounds for matchings and rainbow matchings in product hypergraphs with explicit size conditions, extending classical extremal combinatorics results.

Findings

Bound on the size of families with matching number at most s.

Existence of a family with size bounded by a specific maximum under rainbow matching free conditions.

Conditions on n_i ensuring the bounds hold.

Abstract

Let $n_1,\dots,n_\ell,k_1,\dots,k_\ell$ be integers and let $V_1,\dots,V_\ell$ be disjoint sets with $|V_i|=n_i$ for $i=1,\dots,\ell$. Define $\sqcup_{i=1}^\ell \binom{V_i}{k_i}$ as the collection of all subsets $F$ of $\cup_{i=1}^\ell V_i$ with $|F\cap V_i| =k_i$ for each $i=1,\dots,\ell$. In this paper, we show that if the matching number of $\mathcal{F}\subseteq \sqcup_{i=1}^\ell \binom{V_i}{k_i}$ is at most $s$ and $n_i\geq 4\ell^2 k_i^2s$ for all $i$, then $|\mathcal{F}| \leq \max_{1\leq i\leq \ell}[\binom{n_i}{k_i}-\binom{n_i-s}{k_i}]\prod_{j\neq i}\binom{n_j}{k_j}$. Let $\mathcal{F}_1,\mathcal{F}_2,\dots,\mathcal{F}_s\subseteq\sqcup_{i=1}^\ell \binom{V_i}{k_i}$ with $n_i\geq 8\ell^2k_i^2s$ for all $i$. We also prove that if $\mathcal{F}_1,\mathcal{F}_2,\dots,\mathcal{F}_s$ are rainbow matching free, then there exists $t$ in $[s]$ such that $|\mathcal{F}_t|\leq \max_{1\leq i\leq \ell}\left[\binom{n_i}{k_i}-\binom{n_i-s+1}{k_i}\right]\prod_{j\neq i}\binom{n_j}{k_j}.$