Satisfying sequences for rainbow partite matchings

TL;DR

This paper investigates conditions under which collections of families in a rainbow matching problem guarantee the existence of such matchings, exploring symmetric and asymmetric restrictions with new theoretical insights.

Contribution

It extends previous work by analyzing asymmetric restrictions and answering open questions about satisfying sequences in rainbow matchings.

Findings

Identifies new regimes where satisfying sequences guarantee rainbow matchings.

Provides bounds and conditions for asymmetric family restrictions.

Employs advanced combinatorial and probabilistic methods to establish results.

Abstract

Let $\mathcal F_1,\ldots, \mathcal F_s\subset [n]^k$ be a collection of $s$ families. In this paper, we address the following question: for which sequences $f_1,\ldots, f_s$ the conditions $|\ff_i|>f_i$ imply that the families contain a rainbow matching, that is, there are pairwise disjoint $F_1\in \ff_1,\ldots F_s\in \ff_s$? We call such sequences {\em satisfying}. Kiselev and the first author verified the conjecture of Aharoni and Howard and showed that $f_1 = \ldots = f_s=(s-1)n^{k-1}$ is satisfying for $s>470$. This is the best possible if the restriction is uniform over all families. However, it turns out that much more can be said about asymmetric restrictions. In this paper, we investigate this question in several regimes and in particular answer the questions asked by Kiselev and Kupavskii. We use a variety of methods, including concentration and anticoncentration results, spread approximations, and Combinatorial Nullstellenzats.