# An extension of the Erdős-Stone theorem

## Abstract

Let Kp(u1, ..., up) be the complete p-partite graph whose ith vertex class has ui vertices (l≤i≤p). We show that the theorem of Erdo{combining double acute accent}s and Stone can be extended as follows. There is an absolute constant α>0 such that, for all r≥1, 0<γ<1 and 0<ε≤1/r, every graph G=Gn of sufficiently large order |G|=n with at least {Mathematical expression} edges contains a Kr+1(s,m,...,m,l), where m=m(n)=[α(1-γ)(log n)/log r], s=s(n)=[α(1-γ)(log n)/rlog(1/ε)], and l= l(n) ⌊αe{open}1+γ/2nγ ⌋. The above result strengthens a sharpening of the Erdo{combining double acute accent}s-Stone theorem due to Bollobás, Erdo{combining double acute accent}s, and Simonovits, which guaranteed the existence of a Kr+1(s,...,s) in G. The strengthening in our result lies in the fact that m above is independent of ε and l can be demanded to be almost the first power of n. A related conjecture extending the Chvátal-Szemerédi sharpening of the Erdo{combining double acute accent}s-Stone theorem is presented. © 1994 Akadémiai Kiadó.

## Publication Title

Combinatorica

## Recommended Citation

Bollobás, B., & Kohayakawa, Y.
(1994). An extension of the Erdős-Stone theorem.* Combinatorica**, 14* (3), 279-286.
https://doi.org/10.1007/BF01212976