Van der Waerden's theorem states that for any positive integers r and k there exists a positive integer N such that if the integers {1, 2, ..., N } are colored, each with one of r different colors, then there are at least k integers in arithmetic progression all of the same color. The smallest such N is the van der Waerden number W ( r , k ). As follow up from the article Heuristic lumped Stirling of mathematics, cardinality, induction, and the axiom of choice at Hilbert's paradox of the Grand Hotel Combat Naps - Success Early On Three Laws, Four Basic Forces and Four types of bonds Van der Waerden test: Named after the Dutch mathematician Bartel Leendert van der Waerden, the Van der Waerden test is a statistical test that k population distribution functions are equal. The Van der Waerden test converts the ranks from a standard Kruskal-Wallis one-way analysis of variance to quantiles of the standard normal distribution (details given below). T...