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1 December 2013 Two extensions of Ramsey’s theorem
David Conlon, Jacob Fox, Benny Sudakov
Duke Math. J. 162(15): 2903-2927 (1 December 2013). DOI: 10.1215/00127094-2382566

Abstract

Ramsey’s theorem, in the version of Erdős and Szekeres, states that every 2-coloring of the edges of the complete graph on {1,2,,n} contains a monochromatic clique of order (1/2)logn. In this article, we consider two well-studied extensions of Ramsey’s theorem. Improving a result of Rödl, we show that there is a constant c>0 such that every 2-coloring of the edges of the complete graph on {2,3,,n} contains a monochromatic clique S for which the sum of 1/logi over all vertices iS is at least clogloglogn. This is tight up to the constant factor c and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every k there is an n such that the following holds: for every permutation π of 1,,k1, every 2-coloring of the edges of the complete graph on {1,2,,n} contains a monochromatic clique a1<<ak with

aπ(1)+1aπ(1)>aπ(2)+1aπ(2)>>aπ(k1)+1aπ(k1).

That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, Hajnal, and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.

Citation

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David Conlon. Jacob Fox. Benny Sudakov. "Two extensions of Ramsey’s theorem." Duke Math. J. 162 (15) 2903 - 2927, 1 December 2013. https://doi.org/10.1215/00127094-2382566

Information

Published: 1 December 2013
First available in Project Euclid: 28 November 2013

zbMATH: 1280.05083
MathSciNet: MR3161307
Digital Object Identifier: 10.1215/00127094-2382566

Subjects:
Primary: 05C55
Secondary: 05D10 , 05D40

Rights: Copyright © 2013 Duke University Press

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Vol.162 • No. 15 • 1 December 2013
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