Hereditarily Extended Properties, Quasi-Random Graphs and not Necessarily Induced Subgraphs

Recently nmch at tent ion has been focused on the theory of quasi-random graph and hypergraph properties. The class of quasi-random graphs is defined by certain equivalent graph properties possessed by random graphs. We shall investigate properties P which do not imply quasi-randomness for sequences (Gn) of graphs on their own, but do imply if they hold not only for the whole graph Gn but also for every sufficiently large subgraph of Gn. Here the properties are strongly connected to counting not necessarily induced subgraphs of a given type, while in a subsequent paper we shall investigate the properties connected with counting induced subgraphs.