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 hypergraphM k [ ℐ ] with the property that if G contains roughly the correct number of copies ofM k [ 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