Stability from graph symmetrization arguments in generalized Turán problems

Abstract

Given graphs $H$ and $F$, $\text{ex}\left(n,H,F\right)$ denotes the largest number of copies of $H$ in $F$-free $n$-vertex graphs. Let $\chi \left(H\right)<\chi \left(F\right)=r+1$. We say that $H$ is F-Turán-stable if the following holds. For any $\epsilon >0$ there exists $\delta >0$ such that if an $n$-vertex $F$-free graph $G$ contains at least $\text{ex}\left(n,H,F\right)-\delta n^{\mid V\left(H\right)\mid }$ copies of $H$, then the edit distance of $G$ and the $r$-partite Turán graph is at most $\epsilon n^2$. We say that $H$ is weakly F-Turán-stable if the same holds with the Turán graph replaced by any complete $r$-partite graph $T$. It is known that such stability implies exact results in several cases. We show that complete multipartite graphs with chromatic number at most $r$ are weakly $K_{r+1}$-Turán-stable. Partly answering a question of Morrison, Nir, Norin, Rzażewski, and Wesolek positively, we show that for every graph $H$, if $r$ is large enough, then $H$ is $K_{r+1}$-Turán-stable. Finally, we prove that if $H$ is bipartite, then it is weakly $C_{2k+1}$-Turán-stable for $k$ large enough.