Conflict-free Hypergraph Matchings and Coverings

TL;DR

This paper develops a method to extend conflict-free hypergraph matchings to cover specific vertex subsets while avoiding additional conflicts, simplifying complex proofs in generalized Ramsey theory.

Contribution

It introduces conditions under which conflict-free matchings can be extended to cover designated vertices, accommodating new conflicts and simplifying proof processes.

Findings

Provides a black box theorem for conflict-free matchings

Enables extension of matchings to cover specific vertex subsets

Simplifies proofs in generalized Ramsey theory

Abstract

Recent work showing the existence of conflict-free almost-perfect hypergraph matchings has found many applications. We show that, assuming certain simple degree and codegree conditions on the hypergraph $ \mathcal{H} $ and the conflicts to be avoided, a conflict-free almost-perfect matching can be extended to one covering all of the vertices in a particular subset of $ V(\mathcal{H}) $, by using an additional set of edges; in particular, we ensure that our matching avoids all of a further set of conflicts, which may consist of both old and new edges. This setup is useful for various applications, and our main theorem provides a black box which encapsulates many long and tedious calculations, massively simplifying the proofs of results in generalised Ramsey theory.