An infinite color analogue of Rado's theorem

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:114

Article

Fox, Jacob

关键词: partition regularity;    Rado's theorem;    Ramsey theory;   

DOI  :  10.1016/j.jcta.2007.02.005

来源: Elsevier

【 摘 要 】

Let R be a subring of the complex numbers and a be a cardinal. A system L of linear homogeneous equations with coefficients in R is called a-regular over R if, for every a-coloring of the nonzero elements of R, there is a monochromatic solution to L in distinct variables. In 1943, Rado classified those finite systems of linear homogeneous equations that are a-regular over R for all positive integers a. For every infinite cardinal a, we classify those finite systems of linear homogeneous equations that are a-regular over R. As a corollary, for every positive integer s, we have 2(aleph 0) > aleph(s) if and only if the equation x(0) + Sx(1) = x(s+2) is aleph(0)-regular over R. This generalizes the case s = 1 due to Erdos. (c) 2007 Elsevier Inc. All rights reserved.

【 授权许可】

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