The Speed of Hereditary Properties of Graphs

Given a property P of graphs, write Pn for the set of graphs with vertex set [n] having property P. The growth or speed of a property P can be discussed in terms of the values of |Pn|. For properties with |Pn|<nn hereditary properties are surprisingly well determined by their speeds. Sharpening results of E. R. Scheinerman and J. Zito (1994, J. Combin. Theory Ser. B61, 16–39), we prove numerous results about the possible functions |Pn| and describe in detail the properties exhibiting each type of growth. We also list minimal properties exhibiting each type of growth.