Generalized Ramsey-Turán Numbers
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917665109442560 |
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| author | Balogh, József Magnan, Van Palmer, Cory |
| author_facet | Balogh, József Magnan, Van Palmer, Cory |
| contents | The Ramsey-Turán problem for $K_p$ asks for the maximum number of edges in an $n$-vertex $K_p$-free graph with independence number $o(n)$. In a natural generalization of the problem, cliques larger than the edge $K_2$ are counted. Let {\bf RT}$(n,\#K_q,K_p,o(n))$ denote the maximum number of copies of $K_q$ in an $n$-vertex $K_p$-free graph with independence number $o(n)$. Balogh, Liu and Sharifzadeh determined the asymptotics of {\bf RT}$(n,\# K_3,K_p,o(n))$. In this paper we will establish the asymptotics for counting copies of $K_4$, $K_5$, and for the case $p \geq 5q$. We also provide a family of counterexamples to a conjecture of Balogh, Liu and Sharifzadeh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01804 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Ramsey-Turán Numbers Balogh, József Magnan, Van Palmer, Cory Combinatorics The Ramsey-Turán problem for $K_p$ asks for the maximum number of edges in an $n$-vertex $K_p$-free graph with independence number $o(n)$. In a natural generalization of the problem, cliques larger than the edge $K_2$ are counted. Let {\bf RT}$(n,\#K_q,K_p,o(n))$ denote the maximum number of copies of $K_q$ in an $n$-vertex $K_p$-free graph with independence number $o(n)$. Balogh, Liu and Sharifzadeh determined the asymptotics of {\bf RT}$(n,\# K_3,K_p,o(n))$. In this paper we will establish the asymptotics for counting copies of $K_4$, $K_5$, and for the case $p \geq 5q$. We also provide a family of counterexamples to a conjecture of Balogh, Liu and Sharifzadeh. |
| title | Generalized Ramsey-Turán Numbers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.01804 |