Supersaturation Problem for the Bowtie

Abstract

The Turán function $ex(n,F)$ denotes the maximal number of edges in an $F$-free graph on $n$ vertices. We consider the function $h_F(n,q)$, the minimal number of copies of $F$ in a graph on $n$ vertices with $ex(n,F)+q$ edges. The value of $h_F(n,q)$ has been extensively studied when $F$ is bipartite or colour-critical. In this paper we investigate the simplest remaining graph $F$, namely, two triangles sharing a vertex, and establish the asymptotic value of $h_F(n,q)$ for $q=o(n^2)$.

23 pages, 1 figure

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• An information theoretic approach to Sidorenko's conjecture
• On the logarithimic calculus and Sidorenko's conjecture
• Ramsey properties and extending partial automorphisms for classes of finite structures