The Effect of Induced Subgraphs on Quasi-Randomness

One of the main questions that arise when studying random and quasi-random structures is which properties P are such that any object that satisfies P "behaves" like a truly random one. In the context of graphs, Chung, Graham, and Wilson call a graph p-quasi-random} if it satisfies a long list of the properties that hold in G(n,p) with high probability, like edge distribution, spectral gap, cut size, and more. Our main result here is that the following holds for any fixed graph H: if the distribution of induced copies of H in a graph G is close (in a well defined way) to the distribution we would expect to have in G(n,p), then G is either p-quasi-random or p'-quasi-random, where p' is the unique non-trivial solution of a certain polynomial equation. We thus infer that having the correct distribution of induced copies of any single graph H is enough to guarantee that a graph has the properties of a random one. The proof techniques we develop here, which combine probabilistic, algebraic and combinatorial tools, may be of independent interest to the study of quasi-random structures.