A note on improved bounds for hypergraph rainbow matching problems

TL;DR

This work extends rainbow-matching bounds from graphs to hypergraphs by introducing and analyzing $g(r,n)$, $g'(r,n)$, $h(r,n)$, and $h'(r,n)$ for $r$-uniform (and $r$-partite) hypergraphs. The authors prove a new lower bound $g'(r,n) \ge \frac{2n}{r+1}-\frac{\binom{2r}{r}}{r+1}$, and a near-tight upper bound $g(r,n) \le n-\frac{1}{12r}n^{\frac{r-1}{r}}$ for sufficiently large $n$, revealing a qualitative shift from the $r=2$ case. They also establish corresponding bounds for the related parameters $h(r,n)$ and $h'(r,n)$ using a sampling approach, giving $h(r,n) \ge n+\frac{1}{12r}n^{\frac{r-1}{r}}$ and $h'(r,n) \le \frac{(r+1)n}{2}+3r^2n^{\frac{2r-1}{2r}}$. The results rely on a short double-counting argument, Bollobás’s cross-intersecting theorem, and the Pohoata–Sauermann–Zakharov construction, complemented by a sampling trick to handle strong bounds. The paper also discusses asymptotic conjectures suggesting these rainbow-matching parameters approach $n$ as $n$ grows, and highlights connections to related extremal problems in hypergraphs.

Abstract

A natural question, inspired by the famous Ryser-Brualdi-Stein Conjecture, is to determine the largest positive integer $g(r,n)$ such that every collection of $n$ matchings, each of size $n$, in an $r$-partite $r$-uniform hypergraph contains a rainbow matching of size $g(r,n)$. The parameter $g'(r,n)$ is defined identically with the exception that the host hypergraph is not required to be $r$-partite. In this note, we improve the best known lower bounds on $g'(r,n)$ for all $r \geq 4$ and the upper bounds on $g(r,n)$ for all $r \geq 3$, provided $n$ is sufficiently large. More precisely, we show that if $r\ge3$ then $$\frac{2n}{r+1}-Θ_r(1)\le g'(r,n)\le g(r,n)\le n-Θ_r(n^{1-\frac{1}{r}}).$$ Interestingly, while it has been conjectured that $g(2,n)=g'(2,n)=n-1$, our results show that if $r\ge3$ then $g(r,n)$ and $g'(r,n)$ are bounded away from $n$ by a function which grows in $n$. We also prove analogous bounds for the related problem where we are interested in the smallest size $s$ for which any collection of $n$ matchings of size $s$ in an ($r$-partite) $r$-uniform hypergraph contains a rainbow matching of size $n$.

Theorems & Definitions (27)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Conjecture 1.4
• Definition 1.5: $g(r,n)$, $g'(r,n)$
• Theorem 1.6
• Theorem 1.7
• Definition 1.8: $h(r,n)$, $h'(r,n)$
• Theorem 1.9
• Lemma 2.1