Improved Ramsey bounds for generalized Schur equations

Abstract

We show that for $m,r\in \mathbb{N}$ and $N>(2m+1{)}^r(r!{)}^{1/m}$, every $r$-coloring of the integers in the interval $[N]$ contains a monochromatic solution to the equation \[ x_1 + \dots + \dots x_{m+1} = y_1 + \dots + y_m. \] This generalizes and improves recent results of Koścuiszko. We also show that if $N\ge 2^r$, then every $r$-coloring of the integers in $[N]$ must always determine a monochromatic solution to the above equation for some $m\ge 1$. The latter estimate is optimal.

11 pages