Off-diagonal generalized Schur numbers

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 Sigma (k-1)(i=I) x(i) = x(k) consisting of only red integers, or a solution to Sigma (l-1)(i=1) x(i) = x(l) consisting of only blue integers. We show that S(3, l) = 3l - 4 for odd l greater than or equal to 3, S(3, l) = 3l - 5 for even l greater than or equal to 4, and S(k, l) = kl - l - 1 for 4 less than or equal to k less than or equal to l. (C) 2001 Academic Press.