Stability from graph symmetrization arguments in generalized Turán problems

Abstract

AbstractGiven graphs and , denotes the largest number of copies of in ‐free ‐vertex graphs. Let . We say that is F‐Turán‐stable if the following holds. For any there exists such that if an ‐vertex ‐free graph contains at least copies of , then the edit distance of and the ‐partite Turán graph is at most . We say that is weakly F‐Turán‐stable if the same holds with the Turán graph replaced by any complete ‐partite graph . It is known that such stability implies exact results in several cases. We show that complete multipartite graphs with chromatic number at most are weakly ‐Turán‐stable. Partly answering a question of Morrison, Nir, Norin, Rzażewski, and Wesolek positively, we show that for every graph , if is large enough, then is ‐Turán‐stable. Finally, we prove that if is bipartite, then it is weakly ‐Turán‐stable for large enough.