How many colors guarantee a rainbow matching?

TL;DR

This paper investigates the minimum number of colors needed to guarantee a rainbow matching in multi-hypergraphs, providing improved bounds that are polynomial in the size of the matching for fixed hypergraph uniformity.

Contribution

The authors improve existing upper bounds on the number of colors needed for rainbow matchings, extending results to non-partite hypergraphs and achieving polynomial bounds in t.

Findings

Established new polynomial upper bounds in t for fixed r.

Extended results to non-partite hypergraphs.

Improved upon previous superexponential bounds.

Abstract

Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow $t$-matching in an r-partite r-uniform multi-hypergraph whose edges are colored with f colors such that every color class is a matching of size t. This problem was posed by Aharoni and Berger, who asked to determine the minimum number of colors which guarantees a rainbow matching. We improve on the known upper bounds for this problem for all values of the parameters. In particular for every fixed r, we give an upper bound which is polynomial in t, improving the superexponential estimate of Alon. Our proof also works in the setting not requiring the hypergraph to be r-partite.