Generalized Ramsey–Turán density for cliques
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 Erdős, Sós, Bollobás, Hajnal,...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000295/type/journal_article |
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| Summary: | 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 Erdős, Sós, Bollobás, 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. |
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| ISSN: | 2050-5094 |