On a Generalization of the Ryser-Brualdi-Stein Conjecture

TL;DR

This paper organizes various conjectures related to rainbow matchings in hypergraphs and presents partial results on a central conjecture, advancing understanding of hypergraph matchings and their properties.

Contribution

It introduces a unified framework for multiple conjectures on rainbow matchings and provides new partial results on a key conjecture in the field.

Findings

Multiple conjectures related to rainbow matchings are systematically organized.

Partial results are obtained for a central conjecture on rainbow matchings.

The paper clarifies the relationships among various conjectures in hypergraph theory.

Abstract

A rainbow matching for (not necessarily distinct) sets F_1,...,F_k of hypergraph edges is a matching consisting of k edges, one from each F_i. The aim of the paper is twofold - to put order in the multitude of conjectures that relate to this concept (some of them first presented here), and to present some partial results on one of these conjectures, that seems central among them.