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 $x_1+\cdots +x_{m+1}=y_1+\cdots +y_m$ provided that $N\ge 3^r(r!{)}^{1/m}$, which is qualitatively optimal when $m$ is logarithmic in $r$.

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