## Duke Mathematical Journal

### Two extensions of Ramsey’s theorem

#### Abstract

Ramsey’s theorem, in the version of Erdős and Szekeres, states that every $2$-coloring of the edges of the complete graph on $\{1,2,\ldots,n\}$ contains a monochromatic clique of order $({1}/{2})\log n$. 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 $c\gt 0$ such that every $2$-coloring of the edges of the complete graph on $\{2,3,\ldots,n\}$ contains a monochromatic clique $S$ for which the sum of $1/\log i$ over all vertices $i\in S$ is at least $c\log\log\log n$. 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 $\pi$ of $1,\ldots,k-1$, every $2$-coloring of the edges of the complete graph on $\{1,2,\ldots,n\}$ contains a monochromatic clique $a_{1}\lt \cdots\lt a_{k}$ with

$$a_{\pi(1)+1}-a_{\pi(1)}\gt a_{\pi(2)+1}-a_{\pi(2)}\gt \cdots\gt a_{\pi(k-1)+1}-a_{\pi(k-1)}.$$

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

https://projecteuclid.org/euclid.dmj/1385661571

Digital Object Identifier
doi:10.1215/00127094-2382566

Mathematical Reviews number (MathSciNet)
MR3161307

Zentralblatt MATH identifier
1280.05083

#### 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