Large minimal sets which force long arithmetic progressions

Abstract A 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).