On the degree of regularity of some equations

Abstract

n In this paper we investigate the behaviour of the solutions of equations ~i=t aix, = b, where ~--~=~ ai = 0 and b # 0, with respect to colorings of the set N of positive integers. It tunas out that for any b # 0 there exists an 8-coloring of N, admitting no monochromatic solution of x3 -x2 = x2 x l + 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 xj < x2 < x3, then there exists already a 4-coloring of ~1, which prevents a monochromatic solution of x3 x2 = x2 xl + b, where b E ~.

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