The asymptotic number of triple systems not containing a fixed one Original Research Article

For a fixed 3-uniform hypergraph F, call a hypergraph F-free if it contains no subhypergraph isomorphic to F. Let ex(n,F) denote the size of a largest F-free hypergraph G⊆[n]3. Let Fn(F) denote the number of distinct labelled F-free G⊆[n]3. We show that Fn(F)=2ex(n,F)+o(n3), and discuss related problems.