  For positive integers $s$ and $k_1, k_2, ..., k_s$, let $w(k_1,k_2,...,k_s)$
be the minimum integer $n$ such that any $s$-coloring $\{1,2,...,n\} \to
\{1,2,...,s\}$ admits a $k_i$-term arithmetic progression of color $i$ for some
$i$, $1 \leq i \leq s$. In the case when $k_1=k_2=...=k_s=k$ we simply write
$w(k;s)$.
  That such a minimum integer exists follows from van der Waerden's theorem on
arithmetic progressions.
  In the present paper we give a lower bound for $w(k,m)$ for each fixed $m$.
We include a table with values of $w(k,3)$ which match this lower bound closely
for $5 \leq k \leq 16$. We also give an upper bound for $w(k,4)$, an upper
bound for $w(4;s)$, and a lower bound for $w(k;s)$ for an arbitrary fixed $k$.
We discuss a number of other functions that are closely related to the van der
Waerden function.
