 Solution

The Strengthened Van der Waerden theorem says that every finite coloring contains, for each prescribed $m$, a monochromatic setWe prove the finite form by induction on the number $k$ of colors. The case $k=1$ is immediate. Let $n$ work for $m$ and $k-1$ colors, and apply the ordinary Van der Waerden theorem to obtain a monochromatic progressionIf one of $d,2d,\dots ,nd$ has the progression's color, say $rd$, thenworks. Otherwise $d,2d,\dots ,nd$ use at most $k-1$ colors. By the induction hypothesis their indices contain $b,b+r,\dots ,b+(m-1)r$ together with $r$ in one color. Multiplying by $d$ yieldswhich is the required progression together with its common difference.

Solved by gpt-5.6-sol high.