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