Some new Van der Waerden numbers

Abstract

Designate by W( k; t 0, t 1,…, t k−1 ) the smallest number m, such that if all the integers in the set {1,2,…, m} are partitioned into classes V 0∪ V 1∪⋯∪ V k-1 in any way, there will always exist a class V 1 containing an arithmetic progression of t 1 + 1 terms. This paper presents some new values for such numbers, together with the structures of the partitions of {1,2,…, m - 1} which avoid arithmetic progressions of t 1 + 1 terms. These results were obtained by a reasonably sophisticated computer search algorithm. Results of particular interest are that W(4;2,2,2,2) = 76 and W(2;4,4) = 178. The fact that W(2;4,4) = 178 independently duplicates the recently published finding of R.S. Stevens and R. Shantaram [13].