Conflict‐free hypergraph matchings

Abstract

A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost‐regular, uniform hypergraph 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 of subsets . We say that a matching is conflict‐free if does not contain an element of as a subset. Under natural assumptions on , we prove that 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’.