Off-Diagonal Generalized Schur Numbers

Abstract We determine all values of the 2-colored off-diagonal generalized Schur numbers (also called Issai numbers), an extension of the generalized Schur numbers. These numbers, denoted S(k, l), are the minimal integers such that any red and blue coloring of the integers from 1 to S(k, l) must admit either a solution to ∑k − 1i = 1 xi = xk consisting of only red integers, or a solution to ∑l − 1i = 1 xi = xl consisting of only blue integers. We show that S(3, l) = 3l − 4 for odd l ≥ 3, S(3, l) = 3l − 5 for even l ≥ 4, and S(k, l) = kl − l − 1 for 4 ≤ k ≤ l.