On the van der Waerden numbers w(2;3,t)

Details On the van der Waerden numbers w(2;3,t) On the van der Waerden numbers w(2;3,t)

2011-02-26

arxiv.org/abs/1102.5433v2

Mathematics

Combinatorics

25 pages. Updates of numerical data, improved formulations, and extended discussions on SAT

Scientific paper

We present results and conjectures on the van der Waerden numbers w(2;3,t). We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds. We introduce *palindromic van der Waerden numbers* w^pd(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers w(k; t_0,...,t_{k-1}) however only allowing palindromic solutions (good partitions), defined as reading the same from both ends. Since the underlying property is non-monotonic, these numbers are actually pairs of numbers. We compute w^pd(2;3,t) for 3 <= t <= 25, and we provide lower bounds, which we conjecture to be exact, for 26 <= t <= 39. All computations are based on SAT solving, and we discuss the various relations between SAT solving and Ramsey theory.

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