Generalized Ramsey--Turán density for cliques
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911803620982784 |
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| author | Gao, Jun Jiang, Suyun Liu, Hong Sankar, Maya |
| author_facet | Gao, Jun Jiang, Suyun Liu, Hong Sankar, Maya |
| contents | We study the generalized Ramsey--Turán function $\mathrm{RT}(n,K_s,K_t,o(n))$, which is the maximum possible number of copies of $K_s$ in an $n$-vertex $K_t$-free graph with independence number $o(n)$. The case when $s=2$ was settled by Erd{ő}s, S{ó}s, Bollob{á}s, Hajnal, and Szemerédi in the 1980s. We combinatorially resolve the general case for all $s\ge 3$, showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when $t$ is much larger than $s$. Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12919 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Ramsey--Turán density for cliques Gao, Jun Jiang, Suyun Liu, Hong Sankar, Maya Combinatorics We study the generalized Ramsey--Turán function $\mathrm{RT}(n,K_s,K_t,o(n))$, which is the maximum possible number of copies of $K_s$ in an $n$-vertex $K_t$-free graph with independence number $o(n)$. The case when $s=2$ was settled by Erd{ő}s, S{ó}s, Bollob{á}s, Hajnal, and Szemerédi in the 1980s. We combinatorially resolve the general case for all $s\ge 3$, showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when $t$ is much larger than $s$. Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold. |
| title | Generalized Ramsey--Turán density for cliques |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.12919 |