Ramsey functions related to the van der Waerden numbers (Q1193448)

Van der Waerden's problem was to find to each positive integer \(n\) an interval \(S=[1,\dots,m]\) of positive integers such that if \(S\) is 2- colored, then there exists a monochromatic arithmetic progression of length \(n\) in \(S\), this means, if \(S\) is partitioned into two sets, at least one of these sets contains an arithmetic progression of length \(n\). The least \(m\) with this property is denoted by \(w(n)\). The known upper bounds of \(w(n)\) are very large. The author wants to find smaller upper bounds. The idea is to consider classes \(A'\) of sequences which include the \(n\)- term arithmetic progressions. Then he defines \(w'(n)\) to be the least positive integer which guarantees that if the interval \([1,\dots,w'(n)]\) is 2-colored, then there exists a monochromatic member of \(A'\). For various \(A'\) upper bounds are given for the corresponding \(w'(n)\). It is clear that \(w'(n)\leq w(n)\). In addition, it is shown that the existence of somewhat stronger upper bounds on \(w'(n)\) would imply similar bounds for \(w(n)\).