# van der Waerden

## Concavity of van der Waerden numbers ★★

Author(s): Landman

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}

Keywords: arithmetic progression; van der Waerden