Improved Ramsey bounds for generalized Schur equations

We show that for $m,r\in \mathbb{N}$ and $N>(2m+1{)}^r(r!{)}^{1\mathrm{/}m}$, every $r$-coloring of the integers in the interval $[N]$ contains a monochromatic solution to the equation $$x_1+\cdots +\dots x_{m+1}=y_1+\cdots +y_m\mathrm{.}$$ 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.