Supersaturation Problem for the Bowtie

The Turán function ex(n, F) denotes the maximal number of edges in an F-free graph on n vertices. However if e>ex(n,F), many copies of F appear. We study the function hF(n, q), the minimal number of copies of F in a graph on n vertices with ex(n, F) + q edges. The value of hF(n, q) has been extensively studied when F is colour critical. In this paper we consider a simple non-colour-critical graph, namely the bowtie and establish bounds on hF (n, q) for different ranges of q.