arXiv · 2608.03661
Refined upper bounds on Schur-like numbers
Abstract
For positive integers $r, m$ and $N$, every $r$-coloring of $\{1, \dots, N\}$ contains a monochromatic solution to $x_1+\dots+x_{m+1}=y_1+\dots+y_m$ provided that $N \ge 3^r (r!)^{1/m}$, which is qualitatively optimal when $m$ is logarithmic in $r$.
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Swaroop Hegde, Andrew Lott, Giorgis Petridis, Nagendar Reddy Ponagandla. 2026-08-04. Refined upper bounds on Schur-like numbers. https://arxiv.org/abs/2608.03661
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