Clique Supersaturation

Abstract

We study how many copies of a graph $F$ that another graph $G$ with a given number of cliques is guaranteed to have. For example, one of our main results states that for all $t\ge 2$, if $G$ is an $n$ vertex graph with $kn^{3/2}$ triangles and $k$ is sufficiently large in terms of $t$, then $G$ contains at least \[\Omega(\min\{k^t n^{3/2},k^{\frac{2t^2}{3t-1}}n^{\frac{5t-2}{3t-1}}\})\] copies of $K_{2,t}$, and furthermore, we show these bounds are essentially best-possible provided either $k\ge n^{1/2t}$ or if certain bipartite-analogues of well known conjectures for Tur\'an numbers hold.