Van der Waerden theorem

Every finite coloring of the positive integers contains monochromatic arithmetic progressions of every prescribed finite length.

For all positive integers $k,m$, some $n$ has the following property: every $k$-coloring of $[n]$ contains $a,d>0$ for whichis monochromatic. Thus the common difference has the same color as the progression.

Several monochromatic arithmetic progressions are color-focused when they have different colors and extend by one further term to the same point. Such focused families give an elementary induction proof of the length-three case of the Van der Waerden theorem.