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Given a set of linear equations S with positive integral parameters a₁, , aₖ, k 2, the disjunctive Rado number for the set S is the least positive integer R=Rd (S), if it exists, such that every 2-coloring of the integers in 1, R admits a monochromatic solution to at least one equation in S. We give conditions for the existence of Rd (S), and also give general upper and lower bounds on Rd (S), when S is a set of additive equations \y=x+a₁, , y=x+aₖ\. We also determine Rd (S) when aᵢ is large enough, or when a₁, , aₖ form an arithmetic or geometric progression. We also give conditions for the existence of Rd (S) when S is a set of multiplicative equations \y=a₁ x, , y=aₖ x\. Further, we give a general search-based algorithm to determine Rd (S) when S is a system of equations in two variables, given an upper bound on Rd (S) and an algorithm to determine solutions to S. This algorithm runs in time O (kaₖ aₖ) for the case of additive equations, which is exponentially better than the brute-force algorithm for the problem.
Dileep et al. (Thu,) studied this question.