Equations resolving a conjecture of Rado on partition regularity

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:117

Article

Alexeev, Boris¹  Tsimerman, Jacob¹ 

[1] Princeton Univ, Dept Math, Princeton, NJ 08544 USA

关键词: Colorings;    Partition regularity;    Ramsey theory;   

DOI  :  10.1016/j.jcta.2009.02.009

来源: Elsevier

【 摘 要 】

A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k - 1)-regular but not k-regular. We prove this conjecture by showing that the equation Sigma(k-1)(i=1) 2(i)/2(i)-1x(i) = (-1+Sigma(k-1)(i=1) 2(i)/2(i)-1)x(0) has this property. This conjecture is part of problem E14 in Richard K. Guy's book Unsolved Problems in Number Theory, where it is attributed to Rado's 1933 thesis, Studien zur Kombinatorik. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free