On the finiteness of some n-color Rado numbers

Abstract

For integers k, n, c with k, n ≥ 1, the n-color Rado number Rk(n, c) is defined to be the least integer N if any, or infinity otherwise, such that for every n-coloring of the set {1, 2, . . . , N}, there exists a monochromatic solution in that set to the linear equation x1 + x2 + · · · + xk + c = xk+1. A recent conjecture of ours states that Rk(n, c) should be finite if and only if every divisor d ≤ n of k−1 also divides c. In this paper, we complete the verification of this conjecture for all k ≤ 7. As a key tool, we first prove a general result concerning the degree of regularity over subsets of Z of some linear Diophantine equations.