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An extension of the Erdős-Stone theorem

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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<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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Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, England

    B. Bollobás

  2. Dept. of Pure Math. and Math. Stat., University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, England

    Y. Kohayakawa

  3. Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 20570, 01452-990, São Paulo, SP, Brazil

    Y. Kohayakawa

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

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  • DOI: https://doi.org/10.1007/BF01212976

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