Improved Ramsey bounds for generalized Schur equations

TL;DR

This paper improves bounds on the size of integer intervals needed to guarantee monochromatic solutions to generalized Schur equations under any coloring, extending previous results and establishing optimal thresholds.

Contribution

It provides new, tighter bounds for Ramsey-type problems related to generalized Schur equations, generalizing and improving recent work by Kościusko.

Findings

Monochromatic solutions exist for N > (2m+1)^r (r!)^{1/m}

For N ≥ 2^r, solutions are guaranteed for some m ≥ 1

The bounds are proven to be optimal

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 Ko\'scuiszko. We also show that if $N \geq 2^{r}$, then every $r$-coloring of the integers in $[N]$ must always determine a monochromatic solution to the above equation for some $m \geq 1$. The latter estimate is optimal.