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- W2055441726 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)." @default.
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- W2055441726 date "1986-07-01" @default.
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- W2055441726 title "Large minimal sets which force long arithmetic progressions" @default.
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- W2055441726 doi "https://doi.org/10.1016/0097-3165(86)90097-x" @default.
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