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.
Categories: math.CO