Exact values and lower bounds on the n-color weak Schur numbers for n=2,3

Abstract

For integers k, n with k, n geqslant 1, the n-color weak Schur numberWhspace{-0.6mm}S_{k}(n) is defined as the least integer N, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution x_{1},dots , x_{k}, x_{k+1} in that interval to the equation: x1+x2+⋯+xk=xk+1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} x_{1}+x_{2}+\\dots +x_{k} =x_{k+1}, \\end{aligned}$$\\end{document}with x_{i} ne x_{j}, when ine j. In this paper, we obtain the exact values of WS_{6}(2)=166, WS_{7}(2)=253, WS_{3}(3)=94 and WS_{4}(3)=259 and we show new lower bounds on n-color weak Schur number Whspace{-0.6mm}S_{k}(n) for n=2,3.