| _version_ | 1866901727684329472 |
|---|---|
| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the deep connection between stability theorems in extremal combinatorics and the hypergraph regularity method. Stability results, exemplified by the Erdős-Simonovits theorem for Turan's problem, assert that near-extremal structures must closely resemble the true extremal configurations. The hypergraph regularity lemma, a powerful generalization of Szemerédi's regularity lemma, provides a structural decomposition of any large hypergraph into a collection of random-like components. We demonstrate how this regularity framework can be systematically applied to establish stability for a wide range of extremal hypergraph problems. The methodology involves translating the near-extremal property of a large hypergraph to its much smaller, weighted cluster hypergraph obtained via the regularity lemma. Extremal results applied to this dense cluster hypergraph reveal its structure, which is then lifted back to the original hypergraph to prove its structural similarity to the extremal family. This approach not only provides unified proofs for existing stability theorems but also offers a powerful pathway to resolving new problems where classical combinatorial methods have been less successful. We discuss the strengths and limitations of this method, particularly the challenge of poor quantitative bounds, and outline future directions in this fertile area of research. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17690379 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Stability for Extremal Graph Problems and Hypergraph Regularity SÉRGIO DE ANDRADE, PAULO This paper explores the deep connection between stability theorems in extremal combinatorics and the hypergraph regularity method. Stability results, exemplified by the Erdős-Simonovits theorem for Turan's problem, assert that near-extremal structures must closely resemble the true extremal configurations. The hypergraph regularity lemma, a powerful generalization of Szemerédi's regularity lemma, provides a structural decomposition of any large hypergraph into a collection of random-like components. We demonstrate how this regularity framework can be systematically applied to establish stability for a wide range of extremal hypergraph problems. The methodology involves translating the near-extremal property of a large hypergraph to its much smaller, weighted cluster hypergraph obtained via the regularity lemma. Extremal results applied to this dense cluster hypergraph reveal its structure, which is then lifted back to the original hypergraph to prove its structural similarity to the extremal family. This approach not only provides unified proofs for existing stability theorems but also offers a powerful pathway to resolving new problems where classical combinatorial methods have been less successful. We discuss the strengths and limitations of this method, particularly the challenge of poor quantitative bounds, and outline future directions in this fertile area of research. |
| title | Stability for Extremal Graph Problems and Hypergraph Regularity |
| url | https://doi.org/10.5281/zenodo.17690379 |