Hereditary quasi-random properties of hypergraphs

Thomason and Chung, Graham and Wilson were the first to investigate systematically some properties of quasi-random graphs. They have stated several quite disparate properties of random-like graphs and established their equivalence.Simonovits and Sós introduced a new hereditary property that is equivalent to the other quasi-random properties. For a small fixed graph F, a graph G on n vertices is said to have the Simonovits-Sós Property SSP if for every set X⊆V(G), the number of labeled copies of F in G[X] is given by 2−e(F)|X|v(F)+o(nv(F)). A graph that satisfies SSP for some non-empty graph F is quasi-random.Our contribution in this paper is a natural extension of the result of Simonovits and Sós to 3-uniform hypergraphs.