An extension of the Erdős-Stone theorem

Abstract

LetK _p(u₁, ..., u_p) 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<ε≤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/ε)], andl= l(n) ⌊αɛ^1+γ/2 n ^γ ⌋. The above result strengthens a sharpening of the Erdős-Stone theorem due to Bollobás, 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 ε andl can be demanded to be almost the first power ofn. A related conjecture extending the Chvátal-Szemerédi sharpening of the Erdős-Stone theorem is presented.