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[This post is authored by Luca Trevisan. – T.]
Notions of “quasirandomness” for graphs and hypergraphs have many applications in combinatorics and computer science. Several past posts by Terry have addressed the role of quasirandom structures in additive combinatorics. A recurring theme is that if an object (a graph, a hypergraph, a subset of integers, …) is quasirandom, then several useful properties can be immediately deduced, but, also, if an object is not quasirandom then it possesses some “structure” than can be usefully exploited in an inductive argument. The contrapositive statements of such randomness-structure dichotomies are that certain simple properties imply quasirandomness. One can view such results as algorithms that can be used to “certify” quasirandomness, and the existence (or, in some cases, conjectural non-existence) of such algorithms has implications in computer science, as I shall discuss in this post.
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