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arXiv · 2605.15147

Improved Ramsey bounds for generalized Schur equations

Rafael Miyazaki·Eion Mulrenin·Cosmin Pohoata·Michael Zheng
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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 $N \geq 2^{r}$, 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.

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Rafael Miyazaki, Eion Mulrenin, Cosmin Pohoata, Michael Zheng. 2026-05-14. Improved Ramsey bounds for generalized Schur equations. https://arxiv.org/abs/2605.15147

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