On the finiteness of some n-color Rado numbers

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.