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 and b1, the disjunctive Rado number for the equations x₁+a=x₂ and x₁+b=x₂ is a+b+1-gcd(a,b) if 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 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.