Generalized Turan number with given size

TL;DR

This paper investigates the maximum number of complete subgraphs in graphs avoiding a certain subgraph, providing new bounds and structural insights for generalized Turán problems with given size.

Contribution

It introduces a new method to bound the number of cliques in F-free graphs based on edge count and subgraph structure, improving existing bounds for generalized Turán numbers.

Findings

Established a lower bound on the size of dense subgraphs with many cliques.

Derived an upper bound for the generalized Turán number using classical Turán numbers.

Provided exact asymptotics for the number of cliques in bipartite Turán graphs.

Abstract

Generalized Tur\'an problem with given size, denoted as $\mathrm{mex}(m,K_r,F)$, determines the maximum number of $K_r$-copies in an $F$-free graph with $m$ edges. We prove that for $r\ge 3$ and $\alpha\in(\frac 2 r,1]$, any graph $G$ with $m$ edges and $\Omega(m^{\frac{\alpha r}{2}})$ $K_r$-copies has a subgraph of order $n_0=\Omega(m^\frac{\alpha}{2})$, which contains $\Omega(n_0^{\frac{i(r-2)\alpha}{(2-\alpha)r-2}})$ $K_i$-copies for each $i = 2, \ldots, r$. This implies an upper bound of $\mathrm{mex}(m, K_r, F)$ when an upper bound of $\mathrm{ex}(n,K_r,F)$ is known. Furthermore, we establish an improved upper bound of $\mathrm{mex}(m, K_r, F)$ by $\mathrm{ex}(n, F)$ and $\min_{v_0 \in V(F)} \mathrm{ex}(n, K_r, F - v_0)$. As a corollary, we show $\mathrm{mex}(m, K_r, K_{s,t}) = \Theta( m^{\frac{rs - \binom{r}{2}}{2s-1}} )$ for $r \geq 3$, $s \geq 2r-2$ and $t \geq (s-1)! + 1$, and obtain non-trivial bounds for other graph classes such as complete $r$-partite graphs and $K_s \vee C_\ell$, etc.