= Solution
Let the required <arithmetic progression> have length $m\geq2$, let $k$ be the number of colors, and take $n=n(m,k)$ from the <Hales-Jewett theorem>. Given any <finite coloring> $\chi$ of $[mn]$, color a <word over an alphabet> $w\in[m]^n$ by $\chi(w_1+\cdots+w_n)$. For a <monochromatic> <combinatorial line>, let $A\ne\varnothing$ be its active coordinate set and let $C$ be the sum of its fixed coordinates. The sums of its $m$ words are
$$
C+|A|,\ C+2|A|,\ldots,\ C+m|A|.
$$
They all lie in $[mn]$, have one color, and form an <arithmetic progression> with strictly positive <common difference> $|A|$. Thus, writing $W(k,m)$ for the finite <Van der Waerden theorem> bound,
$$
\boxed{W(k,m)\leq m\,n(m,k).}
$$
The length-one case is immediate. Restricting a <finite coloring> of the <positive integers> to this finite <integer interval> gives the infinite-domain formulation of the <Van der Waerden theorem> as well.
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