σ-Algebras for Quasirandom Hypergraphs

We examine the correspondence between the various notions of quasirandomness for k-uniform hypergraphs and sigma-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 sigma-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 M-k[I] with the property that if G contains roughly the correct number of copies of M-k[I] then G is quasirandom in the sense of I. This generalizes recent results of Kohayakawa, Nagle, Rodl, and Schacht; Conlon, Han, Person, and Schacht; and Lenz and Mubayi giving this result for some particular notions of quasirandomness. (C) 2016 Wiley Periodicals, Inc.