Some extremal problems for hereditary properties of graphs

Given an infinite hereditary property of graphs P, the principal extremal parameter of P is the value [GRAPHICS] The Erdos-Stone theorem gives pi(P) if P is monotone, but this result does not apply to hereditary P. Thus, one of the results of this note is to establish pi(P) for any hereditary property P. Similar questions are studied for the parameter lambda((p)) (G); defined for every real number p >= 1 and every graph G of order n as [GRAPHICS] It is shown that the limit [GRAPHICS] exists for every hereditary property P. A key result of the note is the equality [GRAPHICS] which holds for all p > 1. In particular, edge extremal problems and spectral extremal problems for graphs are asymptotically equivalent.