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 χ 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 =x+a₁,…,y=x+aₖ\. We also determine Rd(S) when max 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 =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 log aₖ) for the case of additive equations, which is exponentially better than the brute-force algorithm for the problem.
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Dileep et al. (2024) studied this question.