Disjunctive Rado numbers

If L(1) and L(2) are linear equations, then the disjunctive Rado number of the set {L(1), L(2)} is the least integer n, provided that it exists, such that for every 2-coloring of the set {1, 2,..., n} there exists a monochromatic solution to either L(1) or L(2). If such an integer n does not exist, then the disjunctive Rado number is infinite. In this paper, it is shown that for all integers a >= 1 and b >= 1, the disjunctive Rado number for the equations x(1) + a = x(2) and x(1) + b = x(2) is a + b + 1 - gcd(a, b) if gcd(a,b)/a + gcd(a,b)/b is odd and the disjunctive Rado number for these equations is infinite otherwise. It is also shown that for all integers a > 1 and b > 1, the disjunctive Rado number for the equations ax(1) = x(2) and bx(1) = x(2) is c(s+t-1) if there exist natural numbers c, s, and t such that a = c(s) and b = c(t) and s + t is an odd integer and c is the largest such integer, and the disjunctive Rado number for these equations is infinite otherwise. (c) 2005 Elsevier Inc. All rights reserved.