Note on forcing pairs

The notion of forcing pairs is located in the study of quasi-random graphs. Roughly speaking, a pair of graphs (F, F ′) is called forcing if the following holds: Suppose for a sequence of graphs (Gn) there is a p > 0 such that the number of copies of F and the number of copies of F ′ in every graph Gn of the sequence (Gn) is approximately the same as the expected value in the random graph G(n, p), then the sequence of graphs (Gn) is quasi-random in the sense of Chung, Graham and Wilson. We describe a construction which, given any graph F with at least one edge, yields a graph F ′ such that (F, F ′) forms a forcing pair. 1 Supported by FAPESP (Proc. 2010/16526-3). 2 Supported through the Heisenberg-Programme of the Deutsche Forschungsgemeinschaft (DFG Grant SCHA 1263/4-1).