A lower estimate of the cyclic van der Waerden numbers
by
Peter Johnson
Auburn University
Coauthors: Jeffrey Burkert

Suppose that k > 2 and r > 0 are integers. The van der Waerden number W(k, r) is the smallest positive integer N such that for every coloring of [N]= 0, ..., N-1 with r colors, there is a monochromatic k-term arithmetic progression somewhere in [N]. The cyclic van der Waerden number Wc(k, r) is the smallest positive integer M such that for all N = M, M + 1, ... for every coloring of [N] with r colors there is a monochromatic k-term arithmetic progression mod N somewhere in [N]. We use a lemma descending from a Euclidean coloring theorem of Arthur Szlam to obtain a lower bound on Wc(k, r); roughly, it's around p^(log-base-2-of-r), where p is the largest prime not exceeding k. As a lower bound of W(k, r) this is not, as one would expect, better than the best known lower bounds of W(k, r), but it is better for finitely many pairs (k, r).

Date received: April 22, 2009