A unified approach to hypergraph stability

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