An extension of the Erd?s-Stone theorem

Abstract

LetK p(u1, ..., up) be the completep-partite graph whoseith vertex class hasu i vertices (l≤i≤p). We show that the theorem of Erdős and Stone can be extended as follows. There is an absolute constant α>0 such that, for allr≥1, 0<γ<1 and 0<e≤1/r, every graphG=G n of sufficiently large order |G|=n with at least $$\left( {1---\frac{1}{r} + \varepsilon } \right) \left( {_2^n } \right)$$ edges contains aK r+1(s,m,...,m,l), wherem=m(n)=[α(1−γ)(logn)/logr],s=s(n)=[α(1−γ)(logn)/rlog(1/e)], andl= l(n) ⌊αɛ1+γ/2 n γ ⌋. The above result strengthens a sharpening of the Erdős-Stone theorem due to Bollobas, Erdős, and Simonovits, which guaranteed the existence of aK r+1(s,...,s) inG. The strengthening in our result lies in the fact thatm above is independent of e andl can be demanded to be almost the first power ofn. A related conjecture extending the Chvatal-Szemeredi sharpening of the Erdős-Stone theorem is presented.