  For positive integers $r,k_0,k_1,...,k_{r-1},$ the van der Waerden number
$w(k_0,k_1,...,k_{r-1})$ is the least positive integer $n$ such that whenever
$\{1,2,...,n\}$ is partitioned into $r$ sets $S_{0},S_{1},...,S_{r-1}$, there
is some $i$ so that $S_i$ contains a $k_i$-term arithmetic progression. We find
several new exact values of $w(k_0,k_1,...,k_{r-1})$. In addition, for the
situation in which only one value of $k_i$ differs from 2, we give a precise
formula for the van der Waerden function (provided this one value of $k_i$ is
not too small)
