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 Turán-type function concerning forbidden induced sub-hypergraphs. This result complements several existing theorems for hereditary and monotone graph and hypergraph properties.