Equations resolving a conjecture of Rado on partition regularity

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 ∑ i = 1 k − 1 2 i 2 i − 1 x i = ( − 1 + ∑ i = 1 k − 1 2 i 2 i − 1 ) x 0 has this property. onjecture is part of problem E14 in Richard K. Guyʹs book “Unsolved Problems in Number Theory”, where it is attributed to Radoʹs 1933 thesis, “Studien zur Kombinatorik”.