On Radoʹs Boundedness Conjecture

We prove that Radoʹs Boundedness Conjecture from Richard Radoʹs 1933 famous dissertation Studien zur Kombinatorik is true if it is true for homogeneous equations. We then prove the first nontrivial case of Radoʹs Boundedness Conjecture: if a 1 , a 2 , and a 3 are integers, and if for every 24-coloring of the positive integers (or even the nonzero rational numbers) there is a monochromatic solution to the equation a 1 x 1 + a 2 x 2 + a 3 x 3 = 0 , then for every finite coloring of the positive integers there is a monochromatic solution to a 1 x 1 + a 2 x 2 + a 3 x 3 = 0 .