A unified approach to hypergraph stability

TL;DR

This paper introduces a unified framework for hypergraph stability theorems, simplifying proofs and strengthening results by reducing stability to checking vertex-extendability in hypergraphs with large minimum degree.

Contribution

The authors develop a general method that simplifies proving hypergraph stability results and provides stronger minimum degree stability conclusions than previous approaches.

Findings

New short proofs of existing hypergraph stability theorems

Establishment of minimum degree stability as a universal approach

Clarification of different stability notions in hypergraph theory

Abstract

We present a method which provides a unified framework for most stability theorems that have been proved in graph and hypergraph theory. Our main result reduces stability for a large class of hypergraph problems to the simpler question of checking that a hypergraph $\mathcal H$ with large minimum degree that omits the forbidden structures is vertex-extendable. This means that if $v$ is a vertex of $\mathcal H$ and ${\mathcal H} -v$ is a subgraph of the extremal configuration(s), then $\mathcal H$ is also a subgraph of the extremal configuration(s). In many cases vertex-extendability is quite easy to verify. We illustrate our approach by giving new short proofs of hypergraph stability results of Pikhurko, Hefetz-Keevash, Brandt-Irwin-Jiang, Bene Watts-Norin-Yepremyan and others. Since our method always yields minimum degree stability, which is the strongest form of stability, in some of these cases our stability results are stronger than what was known earlier. Along the way, we clarify the different notions of stability that have been previously studied.