On Maxima of Takagi-van der Waerden Functions
نویسندگان
چکیده
منابع مشابه
Bounds on Van Der Waerden Numbers and Some Related Functions
For positive integers s and k1, k2, . . . , ks, let w(k1, k2, . . . , ks) be the minimum integer n such that any s-coloring {1, 2, . . . , n} → {1, 2, . . . , s} admits a ki-term arithmetic progression of color i for some i, 1 ≤ i ≤ s. In the case when k1 = k2 = · · · = ks = k we simply write w(k; s). That such a minimum integer exists follows from van der Waerden’s theorem on arithmetic progre...
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For positive integers s and k1,k2, . . . ,ks, the van der Waerden number w(k1,k2, . . . ,ks;s) is the minimum integer n such that for every s-coloring of set {1,2, . . . ,n}, with colors 1,2, . . . ,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this l...
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The classical van der Waerden Theorem says that for every every finite set S of natural numbers and every k-coloring of the natural numbers, there is a monochromatic set of the form aS+b for some a > 0 and b ≥ 0. I.e., monochromatism is obtained by a dilation followed by a translation. We investigate the effect of reversing the order of dilation and translation. S has the variant van der Waerde...
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1 pi − 2 j+1 > 7pr − 4pr > pr . If r is odd, set nk = ∏r1 pi − 2k+1. Then nk ≡ 3(4) (since 32 ≡ 1(4)). But no pi divides nk for 1 ≤ i ≤ r , so the integer nk has some prime factor qk ≡ 3(4) with qk > pr . If j = k, say j > k, the assumption that qk also divides n j leads to the same contradiction as earlier: since nk − n j = 2 j+1 − 2k+1 = 2k+1(2 j−k − 1), we have qk | 2 j−k − 1 and hence qk < ...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1984
ISSN: 0002-9939
DOI: 10.2307/2045305