Strong stability from vertex-extendability and applications in generalized Tur\'an problems
Abstract
Extending the work of Liu--Mubayi--Reiher~LMR23unif on hypergraph Tur\'an problems, we introduce the notion of vertex-extendability for general extremal problems on hypergraphs and develop an axiomatized framework for proving strong stability for extremal problems satisfying certain properties. This framework simplifies the typically complex and tedious process of obtaining stability and exact results for extremal problems into a much simpler task of verifying their vertex-extendability. We present several applications of this method in generalized Tur\'an problems including the Erdos Pentagon Problem, hypergraph Tur\'an-goodness, and generalized Tur\'an problems of hypergraphs whose shadow is complete multipartite. These results significantly strengthen and extend previous results of Erdos~Erdos62, Gyori--J\'anos--Simonovits~GPS91, Grzesik~Gre12, Hatami--Hladk\'y--Kr\'al--Norine--Razborov~HHKNR13, Morrison--Nir--Norin--Rza\.zewski--Wesolek~MNNRPW23, Gerbner--Palmer~GP22, and others.
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