Lower bounds and exact values of the 2-color off-diagonal generalized weak Schur numbers WS(2;k1,k2) (Brief Announcement)

Abstract

In this study, we focus on the concept of the 2-color off-diagonal generalized weak Schur numbers, denoted as WS(2; k1, k2). These numbers are defined for integers ki≥ 2, where i = 1, 2, as the smallest integer M, such that any 2-coloring of the integer interval [1, M] must contain a 2-colored solution to the equation Ekj: x1+x2+ ... +xkj=xkj+1 for j= 1,2, with the condition that xi≠xj when i ≠j. Our objective is to determine lower bounds for these 2-color off-diagonal generalized weak Schur numbers and demonstrate that in several cases, these lower bounds match the exact values.