Large Minimal Sets which Force Long Arithmetic Progressions

A classic theorem of vanderWaerden asserts that for any positive integer k, there is an integer W(k) with the property that if W >- W(k) and the set {1, 2,...,W} is partitioned into r classes C sub 1, C sub 2,...C sub r, then some C will always contain a k-term arithmetic progression. Let us abbreviate this assertion by saying that {1, 2,..., W} arrows AP(k) (written {1, 2,..., W} -> AP(k)). Further, we say that a set X critically arrows AP(k)if:(i) arrows AP(k);(ii) for any proper subset X' is a subset of X, X' does not arrow AP(k).