# Concavity of van der Waerden numbers

 Importance: Medium ✭✭
 Author(s): Landman, Bruce M.
 Subject: Combinatorics » Ramsey Theory
 Keywords: arithmetic progression van der Waerden
 Recomm. for undergrads: no
 Posted by: Bruce Landman on: June 21st, 2007

For $k$ and $\ell$ positive integers, the (mixed) van der Waerden number $w(k,\ell)$ is the least positive integer $n$ such that every (red-blue)-coloring of $[1,n]$ admits either a $k$-term red arithmetic progression or an $\ell$-term blue arithmetic progression.

\begin{conjecture} For all $k$ and $\ell$ with $k \geq \ell$, $w(k,\ell) \geq w(k+1,\ell-1)$. \end{conjecture}

The conjecture was stated in 2000 and published 2003 [LR] and 2007 [KL].

## Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries) *[BL] Bruce Landman and Aaron Robertson, \emph{Ramsey Theory on the Integers}, American Mathematical Society, Providence, Rhode Island, 2003.

[KL] Abdollah Khodkar and Bruce Landman, Recent progress in Ramsey theory on the integers, in \emph{Combinatorial Number Theory}, 305-313, de Gruyter, Berlin, 2007.

* indicates original appearance(s) of problem.