  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".
