Generalized Tur\'an results for intersecting cliques

TL;DR

This paper determines the maximum number of certain subgraphs in large graphs that avoid specific intersecting clique configurations, extending Turán-type extremal results for complex forbidden subgraphs.

Contribution

It provides exact extremal numbers for copies of complete and bipartite graphs in large graphs avoiding intersecting clique structures $B_{r,s}$, generalizing previous Turán results.

Findings

Exact formulas for $ex(n,K_t,B_{r,0})$ and $ex(n,K_t,B_{r,1})$

Determination of $ex(n,K_{a,b},B_{3,1})$ for all parameters

Results hold for sufficiently large $n$

Abstract

For fixed graphs $F$ and $H$, the generalized Tur\'an problem asks for the maximum number $ex(n,H,F)$ of copies of $H$ that an $n$-vertex $F$-free graph can have. In this paper, we focus on cases with $F$ being $B_{r,s}$, the graph consisting of two cliques of size $s$ sharing $r$ common vertices. We determine $ex(n,K_t,B_{r,0})$, $ex(n,K_t,B_{r,1})$ and $ex(n,K_{a,b},B_{3,1})$ for all values of $a,b,r,t$ if $n$ is large enough.