Experimental Mathematics

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

Michal Kouril and Jerome L. Paul

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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.

Article information

Source
Experiment. Math., Volume 17, Issue 1 (2008), 53-61.

Dates
First available in Project Euclid: 18 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.em/1227031896

Mathematical Reviews number (MathSciNet)
MR2410115

Zentralblatt MATH identifier
1151.05048

Subjects
Primary: 68R05: Combinatorics 05D10: Ramsey theory [See also 05C55]

Keywords
Van der Waerden numbers combinatorics high-performance computing Beowulf clusters FPGAs

Citation

Kouril, Michal; Paul, Jerome L. The van der Waerden Number $W(2,6)$ Is 1132. Experiment. Math. 17 (2008), no. 1, 53--61. https://projecteuclid.org/euclid.em/1227031896


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