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 (Combinatorica 9 (1989), 345-362) call a graph p-quasi-random if it satisfies a long list of the properties that hold in G(n,p) with high probability, such as edge distribution, spectral gap, cut size, and so on. 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) over bar)-quasi-random, where (p) over bar is the unique nontrivial solution of the polynomial equation x(delta)(1 - x)(1-delta) = p(delta)(1 - p)(1-delta), with delta being the edge density of H. 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.
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