Monochromatic solutions to systems of exponential equations

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:158

Article

关键词: Ramsey theory;    Arithmetic Ramsey theory;    Ramsey theory on the integers;    Partition regular patterns;    Structure of finite partitions;    Rado's theorem;    Van der Waerden's theorem;   

DOI  :  10.1016/j.jcta.2018.04.002

来源: Elsevier

【 摘 要 】

Let n is an element of N, R be a binary relation on [n], and C-1(i, j), . . . , C-n(i, j) is an element of Z, for i, j is an element of [n]. We define the exponential system of equations epsilon(R, (C-k(i, j)(i,) (j,) (k)) to be the system X-iY1C1(i,X- j) . . .YnCn(i,X- j) = Xj ,X- for (i,X- j) is an element of R,X- in variables X-1, . . . , X-n , Y-1, . . . ,Y-n . The aim of this paper is to classify precisely which of these systems admit a monochromatic solution (X-i , Y-i not equal 1) in an arbitrary finite colouring of the natural numbers. This result could be viewed as an analogue of Rado's theorem for exponential patterns. (C) 2018 Published by Elsevier Inc.

【 授权许可】

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