Supersaturation Problem for the Bowtie

TL;DR

This paper investigates the minimal number of a specific subgraph, two triangles sharing a vertex, in graphs with slightly more edges than the Turán extremal number, focusing on asymptotic behavior when the excess edges are small.

Contribution

It establishes the asymptotic value of the minimal number of such subgraphs for the first non-bipartite, non-critical case when excess edges are small.

Findings

Determined the asymptotic behavior of $h_F(n,q)$ for $q=o(n^2)$.

Extended understanding of supersaturation problems to a new class of graphs.

Provided results for a fundamental non-bipartite graph beyond previously studied cases.

Abstract

The Tur\'an 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 $q=o(n^2)$.