Large minimal sets which force long arithmetic progressions

AbstractA classic theorem of van der Waerden 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 C1, C2,…, Cr, then some Ci 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 Xcritically arrows AP(k) if:(i) X arrows AP(k); (ii) for any proper subset X′ ⊂ X, X′ does not arrow AP(k). The main result of this note shows that for any given k there exist arbitrarily large sets X which critically arrow AP(k)