Hereditary properties of hypergraphs

A hereditary property P^(^k^) is a class of k-graphs closed under isomorphism and taking induced sub-hypergraphs. Let P"n^(^k^) denote those k-graphs of P^(^k^) on vertex set {1,...,n}. We prove an asymptotic formula for log"2|P"n^(^k^)| in terms of a Turan-type function concerning forbidden induced sub-hypergraphs. This result complements several existing theorems for hereditary and monotone graph and hypergraph properties.