Bounds on some van der Waerden numbers

Abstract: For positive integers s and , the van der Waerden number is the minimum integer n such that for every s-coloring of set , with colors , there is a -term arithmetic progression of color i for some i. We give an asymptotic lower bound for for fixed m. We include a table of values of that are very close to this lower bound for . We also give a lower bound for that slightly improves previously-known bounds. Upper bounds for and are also provided. [Copyright &y& Elsevier]