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arXiv · 2403.12919

Generalized Ramsey--Tur\'an density for cliques

Jun Gao·Suyun Jiang·Hong Liu·Maya Sankar
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Abstract

We study the generalized Ramsey--Tur\'an 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{\H{o}}s, S{\'o}s, Bollob{\'a}s, Hajnal, and Szemer\'{e}di 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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Jun Gao, Suyun Jiang, Hong Liu, Maya Sankar. 2024-03-19. Generalized Ramsey--Tur\'an density for cliques. https://arxiv.org/abs/2403.12919

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