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 Pk−1 i=1 2 2i−1xi = “ −1 + Pk−1 i=1 2 2i−1 ” x0 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”. In 1916, Schur [Sch16] proved that in any coloring of the positive integers with finitely many colors, there is a monochromatic solution to x + y = z. In 1927, van der Waerden [vdW27] proved his celebrated theorem that every finite coloring of the positive integers contains arbitrarily long monochromatic arithmetic progressions. In his famous 1933 thesis, Richard Rado [Rad33] generalized these results by classifying the systems of linear equations with monochromatic solutions in every finite coloring. His thesis also contained conjectures regarding equations that do have a finite coloring with no monochromatic solutions. Definition. A linear equation L is k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Remark. Some authors require that the values of the variables be distinct in solutions to L. We follow Rado and Guy in not including this extra condition. Conjecture (Rado, [Rad33] via [Guy04]). For every positive integer k, there exists a linear equation that is (k − 1)-regular but not k-regular. In other words, k is the least number of colors in a coloring of the positive integers with no monochromatic solution to L. Fox and Radoičić [FR05] conjectured that the family of equations Mk given by ∑k−2 i=0 2 xi = 2xk−1 has this property. We prove Rado’s conjecture by using the related family of equations, k−1 ∑ i=1 2 2i − 1 xi = ( −1 + k−1 ∑