Bounds on some van der Waerden numbers

For positive integers s and k1,k2, . . . ,ks, the van der Waerden number w(k1,k2, . . . ,ks;s) is the minimum integer n such that for every s-coloring of set {1,2, . . . ,n}, with colors 1,2, . . . ,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this lower bound for m = 3. We also give a lower bound for w(k,k, . . . ,k;s) that slightly improves previously-known bounds. Upper bounds for w(k,4;2) and w(4,4, . . . ,4;s) are also provided.