Bounds on Van Der Waerden Numbers and Some Related Functions

For positive integers s and k1, k2, . . . , ks, let w(k1, k2, . . . , ks) be the minimum integer n such that any s-coloring {1, 2, . . . , n} → {1, 2, . . . , s} admits a ki-term arithmetic progression of color i for some i, 1 ≤ i ≤ s. In the case when k1 = k2 = · · · = ks = k we simply write w(k; s). That such a minimum integer exists follows from van der Waerden’s theorem on arithmetic progressions. In the present paper we give a lower bound for w(k,m) for each fixed m. We include a table with values of w(k, 3) which match this lower bound closely for 5 ≤ k ≤ 16. We also give an upper bound for w(k, 4), an upper bound for w(4; s), and a lower bound for w(k; s) for an arbitrary fixed k. We discuss a number of other functions that are closely related to the van der Waerden function.