Improved Bounds on Rainbow $k$-partite Matchings

TL;DR

This paper establishes improved bounds on the smallest starting point of arithmetic progressions that guarantee the existence of rainbow matchings in k-partite hypergraphs, extending previous results with new bounds and methods.

Contribution

It provides new lower bounds on satisfying sequences for rainbow matchings in k-partite hypergraphs, including extensions to non-prime n using polynomial methods.

Findings

Established that the sequence is satisfying for c=Ω_k(max(s^2 n^{k-2}, s n^{k-3/2} √log s))

Improved previous bounds by Kupavskii and Popova

Extended results to non-prime n for k=2 using polynomial method

Abstract

Let $n$, $s$, and $k$ be positive integers. We say that a sequence $f_1,\dots,f_s$ of nonnegative integers is satisfying if for any collection of $s$ families $\mathcal F_1,\dots,\mathcal F_s\subseteq [n]^k$ such that $|\mathcal F_i|=f_i$ for all $i$, there exists a rainbow matching, i.e., a list of pairwise disjoint tuples $F_1\in\mathcal F_1$, $\dots$, $F_s\in\mathcal F_s$. We investigate the question, posed by Kupavskii and Popova, of determining the smallest $c=c(n,s,k)$ such that the arithmetic progression $c$, $n^{k-1}+c$, $2n^{k-1}+c$, $\dots$, $(s-1)n^{k-1}+c$ is satisfying. We prove that the sequence is satisfying for $c=\Omega_k(\max(s^2n^{k-2}, sn^{k-3/2}\sqrt{\log s}))$, improving the previous result by Kupavskii and Popova. We also study satisfying sequences for $k=2$ using the polynomial method, extending the previous result by Kupavskii and Popova to when $n$ is not prime.