Disjunctive Rado numbers

If L1 and L2 are linear equations, then the disjunctive Rado number of the set {L1, L2} 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 L1 or L2. 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 x1 + a = x2 and x1 + b = x2 is a + b + 1− gcd(a, b) if a gcd(a,b) + b gcd(a,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 ax1= x2 and bx1= x2 is cs+t−1 if there exist natural numbers c, s, and t such that a= cs and b= ct 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. © 2005 Elsevier Inc. All rights reserved.