A unified approach to hypergraph stability

Abstract: We present a method which provides a unified framework for many 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.

Our method always yields an Andrásfai-Erdős-Sós type result, which says if $\mathcal H$ has large minimum degree, then it must be a subgraph of one of the extremal configurations.

This is joint work with Dhruv Mubayi and Christian Reiher.

combinatoricsprobability

Audience: researchers in the topic

Extremal and probabilistic combinatorics webinar

Series comments: We've added a password: concatenate the 6 first prime numbers (hence obtaining an 8-digit password).