The Generalized Tur\'{a}n Problem of Two Intersecting Cliques

TL;DR

This paper investigates the maximum number of complete subgraphs in graphs avoiding two intersecting cliques, providing exact and asymptotic results for specific cases using symmetrization and structural theorems.

Contribution

It determines the maximum number of triangles and larger cliques in graphs avoiding two intersecting cliques for certain parameters, extending classical Turán-type results.

Findings

Exact maximum for $K_4$ in $B_{4,1}$-free graphs for large $n$

Asymptotic formula for $ex(n,K_r,B_{r,1})$ when $r o ext{large}$

Utilization of symmetrization and structure theorems in extremal graph problems

Abstract

For $s<r$, let $B_{r,s}$ be the graph consisting of two copies of $K_r$, which share exactly $s$ vertices. Denote by $ex(n, K_r, B_{r,s})$ the maximum number of copies of $K_r$ in a $B_{r,s}$-free graph on $n$ vertices. In 1976, Erd\H{o}s and S\'{o}s determined $ex(n,K_3,B_{3,1})$. Recently, Gowers and Janzer showed that $ex(n,K_r,B_{r,r-1})=n^{r-1-o(1)}$. It is a natural question to ask for $ex(n,K_r,B_{r,s})$ for general $r$ and $s$. In this paper, we mainly consider the problem for $s=1$. Utilizing the Zykov's symmetrization, we show that $ex(n,K_4, B_{4,1})=\lfloor (n-2)^2/4\rfloor$ for $n\geq 45$. For $r\geq 5$ and $n$ sufficiently large, by the F\"{u}redi's structure theorem we show that $ex(n,K_r,B_{r,1}) =\mathcal{N}(K_{r-2},T_{r-2}(n-2))$, where $\mathcal{N}(K_{r-2},T_{r-2}(n-2))$ represents the number of copies of $K_{r-2}$ in the $(r-2)$-partite Tur\'{a}n graph on $n-2$ vertices.