The Determination of 2-color Zero-sum Generalized Schur Numbers

Aaron Robertson, Bidisha Roy, Subha Sarkar
Consider the equation E : x 1 + · · · + x k1 = x k and let k and r be positive integers such that r | k. The number S z,2 (k; r) is defined to be the least positive integer t such that for any 2-coloring : [1, t] ! {0, 1} there exists a solution (ˆ x 1 , ˆ x 2 ,. .. , ˆ x k) to equation E satisfying k X i=1 (ˆ x i) ⌘ 0 (mod r). In a recent paper, the first author posed the question of determining the exact value of S z,2 (k; 4). In this article, we solve this problem and show, more generally,
more » ... , more generally, that S z,2 (k, r) = kr 2r + 1 for all positive integers k and r with r | k and k 2r.