Stability from graph symmetrization arguments in generalized Turán problems

Given graphs (Formula presented.) and (Formula presented.), (Formula presented.) denotes the largest number of copies of (Formula presented.) in (Formula presented.) -free (Formula presented.) -vertex graphs. Let (Formula presented.). We say that (Formula presented.) is F-Turán-stable if the following holds. For any (Formula presented.) there exists (Formula presented.) such that if an (Formula presented.) -vertex (Formula presented.) -free graph (Formula presented.) contains at least (Formula presented.) copies of (Formula presented.), then the edit distance of (Formula presented.) and the (Formula presented.) -partite Turán graph is at most (Formula presented.). We say that (Formula presented.) is weakly F-Turán-stable if the same holds with the Turán graph replaced by any complete (Formula presented.) -partite graph (Formula presented.). It is known that such stability implies exact results in several cases. We show that complete multipartite graphs with chromatic number at most (Formula presented.) are weakly (Formula presented.) -Turán-stable. Partly answering a question of Morrison, Nir, Norin, Rzażewski, and Wesolek positively, we show that for every graph (Formula presented.), if (Formula presented.) is large enough, then (Formula presented.) is (Formula presented.) -Turán-stable. Finally, we prove that if (Formula presented.) is bipartite, then it is weakly (Formula presented.) -Turán-stable for (Formula presented.) large enough. © 2024 Wiley Periodicals LLC.