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".

Boris Alexeev and Jacob Tsimerman
*Equations resolving a conjecture of Rado on partition regularity*

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