Random Strategies are Nearly Optimal for Generalized van der Waerden Games
نویسندگان
چکیده
منابع مشابه
On generalized van der Waerden triples
Van der Waerden’s classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k, r), such that any rcoloring of {1, 2, . . . , w(k, r)} must contain a monochromatic k-term arithmetic progression {x, x+d, x+2d, . . . , x+(k−1)d}. We investigate the following generalization of w(3, r). For fixed positive integers a and b wit...
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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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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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In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds. The area of Ramsey theory is especially rich in such results. Using the problem of computing van der Waerden n...
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Our subject is infinitary Ramsey theory, specifically the existence of monochromatic structures of infinite cardinality for finite colorings of semigroups. Specifically, we shall prove an infinitary version of a recent polynomial extension of van der Waerden's theorem due to Bergelson and Leibman ([BL1]). Alternatively, this theorem may be viewed as a polynomial version of an infinitary van der...
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ژورنال
عنوان ژورنال: Electronic Notes in Discrete Mathematics
سال: 2017
ISSN: 1571-0653
DOI: 10.1016/j.endm.2017.07.037