Improved Ramsey bounds for generalized Schur equations

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.