Conflict-free hypergraph matchings

Abstract

A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost-regular, uniform hypergraph $H$ 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 $C$ of subsets $C\subseteq E\left(H\right)$. We say that a matching $M\subseteq E\left(H\right)$ is conflict-free if $M$ does not contain an element of $C$ as a subset. Under natural assumptions on $C$, we prove that $H$ 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’.