 Solution

Let the required arithmetic progression have length $m\ge 2$, 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 areThey 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,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.