Duke Mathematical Journal
- Duke Math. J.
- Volume 162, Number 15 (2013), 2903-2927.
Two extensions of Ramsey’s theorem
Ramsey’s theorem, in the version of Erdős and Szekeres, states that every -coloring of the edges of the complete graph on contains a monochromatic clique of order . 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 such that every -coloring of the edges of the complete graph on contains a monochromatic clique for which the sum of over all vertices is at least . This is tight up to the constant factor and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every there is an such that the following holds: for every permutation of , every -coloring of the edges of the complete graph on contains a monochromatic clique with
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 . We make progress towards this conjecture, obtaining an upper bound on which is exponential in a power of . This improves a result of Shelah, who showed that is at most double-exponential in .
Duke Math. J., Volume 162, Number 15 (2013), 2903-2927.
First available in Project Euclid: 28 November 2013
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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