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arXiv · 2311.15289

Counting cliques without generalized theta graphs

Jun Gao·Zhuo Wu·Yisai Xue
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Abstract

The \textit{generalized Turán number} $\mathrm{ex}(n, T, F)$ is the maximum possible number of copies of $T$ in an $F$-free graph on $n$ vertices for any two graphs $T$ and $F$. For the book graph $B_t$, there is a close connection between $\ex(n,K_3,B_t)$ and the Ruzsa-Szemerédi triangle removal lemma. Motivated by this, in this paper, we study the generalized Turán problem for generalized theta graphs, a natural extension of book graphs. Our main result provides a complete characterization of the magnitude of $\ex(n,K_3,H)$ when $H$ is a generalized theta graph, indicating when it is quadratic, when it is nearly quadratic, and when it is subquadratic. Furthermore, as an application, we obtain the exact value of $\ex(n, K_r, kF)$, where $F$ is an edge-critical generalized theta graph, and $3\le r\le k+1$, extending several recent results.

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Jun Gao, Zhuo Wu, Yisai Xue. 2026-02-24. Counting cliques without generalized theta graphs. https://arxiv.org/abs/2311.15289

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