Some new van der Waerden numbers and some van der Waerden-type numbers

Abstract

The van der Waerden number w.rI k 1 ; k 2 ; : : : ; k r / is the least m such that given any partition ¹1; 2; : : : ; mº D P 1 [ P 2 [ [ P r , there is an index j 2 ¹1; 2; : : : ; rº such that P j contains an arithmetic progression of length k j . We have computed exact values of some (30) previously unknown van der Waerden numbers and also computed lower bounds of others. Let w d .rI k 1 ; k 2 ; : : : ; k r / be the least m such that given any partition ¹1; 2; : : : ; mº D P 1 [ P 2 [ [ P r , there is an index j 2 ¹1; 2; : : : ; r 1º such that P j contains an arithmetic progression of length k j , or P r contains an arithmetic progression of length k r with common difference at most d . A table of observed values of w d .rI k 1 ; k 2 ; : : : ; k r / for d D 1; 2; : : : ; b m 1 kr 1 c is given.

Two New Van Der Waerden Numbers: w(2; 3, 17) and w(2; 3, 18)

“…A list of known van der Waerden numbers was recently published in [1]. In Table 1, we present the exact values of w.2I 3; 17/ and w.2I 3; 18/.…”

“…All of the 30 numbers reported in [1] were computed using this branching rule. In addition, this branching rule helps us to recognize a pattern in the size of the subtrees when we split the DPLL-tree of w.2I 3; k/ for distributed computation, as described in the following section.…”

“…Given any positive integer r and positive integers k 1 ; k 2 ; : : : ; k r , there is an integer m such that given any partition ¹1; 2; : : : ; mº D P 1 [ P 2 [ [ P r (1) there is always a class P j containing an arithmetic progression of length k j . Let us denote the least m with this property by w.rI k 1 ; k 2 ; : : : ; k r /.…”

On Computation of Exact van der Waerden Numbers

“…Van der Waerden's theorem [18] can be formulated (as in Chvátal [5]) as follows: Given any positive integer k and positive integers t 0 ; t 1 ; : : : ; t k 1 , there is an integer m such that given any partition ¹1; 2; : : : ; mº D P 0 [ P 1 [ [ P k 1 (1) there is always a class P j containing an arithmetic progression of length t j . Let us denote the least positive integer m with this property by w.kI t 0 ; t 1 ; : : : ; t k 1 /.…”

“…By a good partition, we mean a partition of the form (1) such that no P j contains an arithmetic progression of t j terms. We have recently published some previously unknown van der Waerden numbers in [1,2]. 418 T.…”