Conflict‐free hypergraph matchings

Abstract

AbstractA 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’.