Bipartite Subgraphs and Quasi-Randomness

We say that a family of graphs G = {Gn : n ≥ 1} is p-quasi-random, 0 < p < 1, if it shares typical properties of the random graph G(n, p); for a definition, see below. We denote by Q(p) the class of all graphs H for which e(Gn) ≥ (1 + o(1))p ( n 2 ) and the number of not necessarily induced labeled copies of H in Gn is at most (1 + o(1))pn imply that G is p-quasi-random. In this note, we show that all complete bipartite graphs Ka,b, a, b ≥ 2, belong to Q(p) for all 0 < p < 1.