Duke Mathematical Journal

Two extensions of Ramsey’s theorem

David Conlon, Jacob Fox, and Benny Sudakov

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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.

Article information

Source
Duke Math. J., Volume 162, Number 15 (2013), 2903-2927.

Dates
First available in Project Euclid: 28 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1385661571

Digital Object Identifier
doi:10.1215/00127094-2382566

Mathematical Reviews number (MathSciNet)
MR3161307

Zentralblatt MATH identifier
1280.05083

Subjects
Primary: 05C55: Generalized Ramsey theory [See also 05D10]
Secondary: 05D10: Ramsey theory [See also 05C55] 05D40: Probabilistic methods

Citation

Conlon, David; Fox, Jacob; Sudakov, Benny. Two extensions of Ramsey’s theorem. Duke Math. J. 162 (2013), no. 15, 2903--2927. doi:10.1215/00127094-2382566. https://projecteuclid.org/euclid.dmj/1385661571


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