Conflict-free hypergraph matchings

TL;DR

This paper extends classical hypergraph matching results to include conflict constraints, introducing a conflict-free matching process with applications to Steiner systems and using a novel random greedy algorithm.

Contribution

It introduces the concept of conflict-free matchings in hypergraphs and proves their existence under certain conditions, extending classical theorems.

Findings

Existence of conflict-free, almost-perfect matchings in hypergraphs with conflict constraints.

Development of a conflict-free matching process using a random greedy algorithm.

Application to asymptotic results for high-girth Steiner systems.

Abstract

A celebrated theorem of Pippenger, and Frankl and R\"odl states that every almost-regular, uniform hypergraph $\mathcal{H}$ with small maximum codegree has an almost-perfect matching. We extend this result by obtaining a ``conflict-free'' matching, where conflicts are encoded via a collection $\mathcal{C}$ of subsets $C\subseteq E(\mathcal{H})$. We say that a matching $\mathcal{M}\subseteq E(\mathcal{H})$ is conflict-free if $\mathcal{M}$ does not contain an element of $\mathcal{C}$ as a subset. Under natural assumptions on $\mathcal{C}$, we prove that $\mathcal{H}$ has a conflict-free, almost-perfect matching. This has many applications, one of which yields new asymptotic results for so-called ``high-girth'' Steiner systems. Our main tool is a random greedy algorithm which we call the ``conflict-free matching process''.