An extension of the Erdős-Stone theorem


LetK p(u1, ..., up) be the completep-partite graph whoseith vertex class hasu i vertices (l≤ip). 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) ⌊αɛ1+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.

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Bollobás, B., Kohayakawa, Y. An extension of the Erdős-Stone theorem. Combinatorica 14, 279–286 (1994).

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AMS subject classification code (1991)

  • 05 C 35