On the degree of regularity of some equations |
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Authors: | Arie Bialostocki Hanno Lefmann Terry Meerdink |
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Affiliation: | a Department of Mathematics and Statistics, University of Idaho, Moscow, Idaho 83843, USA b Universität Dortmund, Fachbereich Informatik, LS II, D-44221, Dortmund, Germany |
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Abstract: | In this paper we investigate the behaviour of the solutions of equations ΣI=1n aixi = b, where Σi=1n, ai = 0 and b ≠ 0, with respect to colorings of the set N of positive integers. It turns out that for any b ≠ 0 there exists an 8-coloring of N, admitting no monochromatic solution of x3 − x2 = x2 − x1 + b. For this equation, for b odd and 2-colorings, only an odd-even coloring prevents a monochromatic solution. For b even and 2-colorings, always monochromatic solutions can be found, and bounds for the corresponding Rado numbers are given. If one imposes the ordering x1 < x2 < x3, then there exists already a 4-coloring of N, which prevents a monochromatic solution of x3 − x2 = x2 − x1 + b, where b ε N. |
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