6 - Dependent Random Choice

If F is a family of graphs containing a bipartite graph, then for some δ > 0, ex(n,F) < n2−δ by the Kövari-Sós-Turán Theorem. Erdős and Simonovits [5] conjectured that for every finite family graph F of graphs, there exists α = α(F) such that ex(n,F) = Θ(nα). This α is called the exponent of F . If F is a bipartite graph containing a cycle, then in general it is not known if F = {F} has an exponent, even for simple graphs such as F = K4,4 and F = C8. In these notes, we give an approach to finding upper bounds on the exponent using a method called Dependent Random Choice, which has many other applications besides. We refer the reader to Fox and Sudakov [6] for a survey of dependent random choice.