(For example, for r = 3, the longest coloring that avoids an arithmetic progression of length 2 is RGB.)
[1] Some lower bound colorings computed using SAT approach by Marijn J.H.
Heule [6] can be found on github project page.
Van der Waerden numbers with r ≥ 2 are bounded above by as proved by Gowers.
Following is a list of some known van der Waerden numbers: w(2; 3,3) 9 Chvátal [2] Van der Waerden numbers are primitive recursive, as proved by Shelah;[23] in fact he proved that they are (at most) on the fifth level