On Computation of Exact van der Waerden Numbers

Abstract

The van der Waerden number w.kI t 0 ; t 2 ; : : : ; t k 1 / is the smallest integer m such that in every partition P 0 [ P 1 [ [ P k 1 of the set ¹1; 2; : : : ; mº there is always a block P j that contains an arithmetic progression of length t j . In this paper, we report the exact value of the previously unknown van der Waerden number w.2I 4; 9/, some lower bounds of w.2I 5; t/ and polynomial upper-bound conjectures for w.2I 4; t / and w.2I 5; t /. We also present an efficient SAT-encoding of the number w.kI t 0 ; t 1 ; : : : ; t k 1 / for k > 3 using which we have computed the exact value of w.3I 3; 3; 6/ and some lower bounds of w.3I t 0 ; t 1 ; t 2 /.