σ‐algebras for quasirandom hypergraphs

We examine the correspondence between the various notions of quasirandomness for k‐uniform hypergraphs and σ‐algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a σ‐algebra defined by a collection of subsets of [1,k]. We associate each notion of quasirandomness ℐ with a collection of hypergraphs, the ℐ‐adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each ℐ‐adapted hypergraph. We then identify, for each ℐ, a particular ℐ‐adapted hypergraph Mk[ℐ] with the property that if G contains roughly the correct number of copies of Mk[ℐ] then G is quasirandom in the sense of ℐ. This generalizes recent results of Kohayakawa, Nagle, Rödl, and Schacht; Conlon, Hàn, Person, and Schacht; and Lenz and Mubayi giving this result for some particular notions of quasirandomness. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 114–139, 2017