Off-Diagonal Generalized Schur Numbers

Authors: Robertson A.¹; Schaal D.²

Source: Advances in Applied Mathematics, Volume 26, Number 3, April 2001 , pp. 252-257(6)

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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 sum ^{k - 1}_{i = 1} x_i = x_k consisting of only red integers, or a solution to sum ^{l - 1}_{i = 1} x_i = x_l 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. Copyright 2001 Academic Press.

Language: English

Document Type: Research article

Affiliations: 1: Department of Mathematics, Colgate University, Hamilton, New York, 11346 2: Department of Mathematics and Statistics, South Dakota State University, Brookings, South Dakota, 57007

Publication date: 2001-04-01