The van der Waerden Number $W(2,6)$ Is 1132

Abstract

We have verified that the van der Waerden number $W(2, 6)$ is 1132, that is, 1132 is the smallest integer n = W(2, 6) such that whenever the set of integers {1, 2, . . . , $n$} is 2-colored, there exists a monochromatic arithmetic progression of length 6. This was accomplished by applying special preprocessing techniques that drastically reduced the required search space. The exhaustive search showing that $W(2, 6)$ = 1132 was carried out by formulating the problem as a satisfiability (SAT) question for a Boolean formula in conjunctive normal form (CNF), and then using a SAT solver specifically designed for the problem. The parallel backtracking computation was run over multiple Beowulf clusters, and in the last phase, field programmable gate arrays (FPGAs) were used to speed up the search. The fact that $W(2, 6)$ > 1131 was shown previously by the first author.