Journal of Graph Theory
A diagonal form for the incidence matrices of t-subsets vs. k-subsets
European Journal of Combinatorics
The algorithmic aspects of the regularity lemma
Journal of Algorithms
Hereditary Extended Properties, Quasi-Random Graphs and Induced Subgraphs
Combinatorics, Probability and Computing
Regularity lemma for k-uniform hypergraphs
Random Structures & Algorithms
The Difficulty of Testing for Isomorphism against a Graph That Is Given in Advance
SIAM Journal on Computing
Quasirandomness, Counting and Regularity for 3-Uniform Hypergraphs
Combinatorics, Probability and Computing
The counting lemma for regular k-uniform hypergraphs
Random Structures & Algorithms
A variant of the hypergraph removal lemma
Journal of Combinatorial Theory Series A
Combinatorics, Probability and Computing
Quasi-random graphs and graph limits
European Journal of Combinatorics
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One of the main questins 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 [9] 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 distribuition we would expect to have in G(n,p), then G is either P-quasi-random or p-quasirandom, where &pmacr; is the unique non-trivial solution of the polynomial equation xδ (1 -- x)1-δ = pδ (1 -- p)1--δ, with δ 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.