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