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 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 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.