Some new exact van der Waerden numbers. (English) Zbl 1110.05097
The generalized van der Waerden number \(w(k_0, \dots, k_{r-1};r)\) is the least positive integer \(n\) such that whenever \(\{1,2, \dots , n \}\) is partioned into \(r\) sets \(S_0,\ldots, S_{r-1}\) there is some \(i\) so that \(S_i\) contains a \(k_i\)-term progression. The authors determine several new exact values and list the number of different colourings in the extremal cases. They also give a general formula for \(w(k,2,2, \dots, 2;r)\) for sufficiently large \(k\).
Reviewer: Christian Elsholtz (Surrey)
Online Encyclopedia of Integer Sequences:
Van der Waerden numbers w(j+2; t_0,t_1,...,t_{j-1}, 3, 3) with t_0 = t_1 = ... = t_{j-1} = 2.Van der Waerden numbers w(j+3; t_0,t_1,...,t_{j-1}, 3, 3, 3) with t_0 = t_1 = ... = t_{j-1} = 2.
Van der Waerden numbers w(j+2; t_0,t_1,...,t_{j-1}, 3, 6) with t_0 = t_1 = ... = t_{j-1} = 2.
Van der Waerden numbers w(3; 3, 3, n).
Van der Waerden numbers w(j+2; t_0,t_1,...,t_{j-1}, 4, 6) with t_0 = t_1 = ... = t_{j-1} = 2.