{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/strong-forms-of-stability-from-flag-algebra","title":"Strong Forms of Stability from Flag Algebra Calculations","arxiv_id":"1706.02612","date":"2017-06-08","proceeding":null,"authors":["Oleg Pikhurko","Jakub Sliacan","Konstantinos Tyros"],"abstract":"Given a hereditary family $\\mathcal{G}$ of admissible graphs and a function $\\lambda(G)$ that linearly depends on the statistics of order-$\\kappa$ subgraphs in a graph $G$, we consider the extremal problem of determining $\\lambda(n,\\mathcal{G})$, the maximum of $\\lambda(G)$ over all admissible graphs $G$ of order $n$. We call the problem perfectly $B$-stable for a graph $B$ if there is a constant $C$ such that every admissible graph $G$ of order $n\\ge C$ can be made into a blow-up of $B$ by changing at most $C(\\lambda(n,\\mathcal{G})-\\lambda(G)){n\\choose2}$ adjacencies. As special cases, this property describes all almost extremal graphs of order $n$ within $o(n^2)$ edges and shows that every extremal graph of order $n\\ge n_0$ is a blow-up of $B$. We develop general methods for establishing stability-type results from flag algebra computations and apply them to concrete examples. In fact, one of our sufficient conditions for perfect stability is stated in a way that allows automatic verification by a computer. This gives a unifying way to obtain computer-assisted proofs of many new results.","url_abs":"http://arxiv.org/abs/1706.02612v2","url_pdf":"http://arxiv.org/pdf/1706.02612v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"strong-forms-of-stability-from-flag-algebra","repo_url":"https://github.com/jsliacan/flagmatic","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}