A general lower bound on the weak Schur number

Abstract

For integers k, n with $k,n\ge 1$, the n-color weak Schur number $W{S}_k\left(n\right)$ 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 $x_1+x_2+\dots +x_k=x_{k+1}$, with $x_i\ne x_j$, when $i\ne j$. We show a relationship between $W{S}_k(n+1)$ and $W{S}_k\left(n\right)$ and a general lower bound on the $W{S}_k\left(n\right)$ is obtained.