Equations resolving a conjecture of Rado on partition regularity

Details Equations resolving a conjecture of Rado on partition regularity Equations resolving a conjecture of Rado on partition regularity

2008-12-08

arxiv.org/abs/0812.1314v2

Journal of Combinatorial Theory, Series A 117 (2010), no. 7, 1008-1010

Mathematics

Combinatorics

2 pages, no figures. The replaced version (v2) differs from the original (v1) only in exposition

Scientific paper

10.1016/j.jcta.2009.02.009

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 $\sum_{i=1}^{k-1} \frac{2^i}{2^i-1} x_i = (-1 + \sum_{i=1}^{k-1} \frac{2^i}{2^i-1}) x_0$ 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".

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