Improved Ramsey Bounds for Generalized Schur Equations

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remark_2_2: Records the (4l-2)^q (q!)^(1/l) + 1 upper bound and its direct but nonresolving relevance to the numerator in Problem 554.

theorem_1_1: Bounds S_m(r) by (2m+1)^r (r!)^(1/m) + 1 for every positive m and r.

theorem_1_3: Every r-coloring of [2^r] forces the generalized Schur equation for some m, and 2^r is the least interval size with this property.

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Rafael Miyazaki, Eion Mulrenin, Cosmin Pohoata, and Michael Zheng, Improved Ramsey Bounds for Generalized Schur Equations, arXiv:2605.15147v1 (14 May 2026). The supplied record establishes this preprint version; it does not establish acceptance or publication. The arXiv record (https://arxiv.org/abs/2605.15147, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Local artifact.

• Selected arXiv v1 PDF, 11 physical pages. Theorem 1.1 is on physical and printed p. 2, and its proof is on pp. 6--7. Theorem 1.3 and the paper's explicit graph/additive distinction are on p. 3; its proof is on pp. 7--9. Lemma 2.1 is on p. 4, and Remark 2.2 is on p. 5.

Version check of 2026-09-17: the arXiv listing still shows only v1 (14 May 2026) and no journal reference, and a Crossref bibliographic query found no publication record; the Ramsey bound of Remark 2.2 therefore rests on an unrefereed preprint, and the page for Problem 554 carries that qualification. Read status: claims checked for Remark 2.2 (p. 5, read clause by clause in the text layer, with the page image rendered, on 2026-09-17); the derivation it delegates to the Axenovich et al. argument was not checked.

For $m,r\in\mathbb N$, let $S_m(r)$ be the least $N$ such that every $r$-coloring of $[N]$ contains a monochromatic solution of

$$x_1+\cdots+x_{m+1}=y_1+\cdots+y_m.$$

Theorem 1.1 proves

$$S_m(r)\leq(2m+1)^r(r!)^{1/m}+1.$$

Theorem 1.3 determines a different threshold: every $r$-coloring of $[2^r]$ has a monochromatic solution for some $m$, and $2^r$ is minimal for that property.

Remark 2.2 records the fixed-cycle graph consequence of the sharpened Lemma 2.1:

$$r(C_{2\ell+1};q)\leq(4\ell-2)^q(q!)^{1/\ell}+1.$$

For every fixed $\ell\geq2$, this directly bounds the numerator in Problem 554 (rename $q$ as its number of colors). It supplies no comparison with $R_q(K_3)$ proving that the ratio tends to zero, so it does not resolve Problem 554.

These are additive-coloring results, not bounds for the shortest monochromatic odd cycle in an $r$-edge-coloring of $K_{2^r+1}$. The paper itself explains why the standard difference coloring does not reverse this gap: an odd monochromatic cycle gives an equality of two monochromatic sums, but their numbers of terms need not differ by exactly one. Thus neither theorem updates Problem 609.

Source: https://arxiv.org/abs/2605.15147.

Bears on. #554 through Remark 2.2's direct but nonresolving numerator bound, and #609 through the additive results as non-transferring context.

Results to transcribe.

• Theorem 1.1: for all $m,r\in\mathbb N$, $S_m(r)\leq(2m+1)^r(r!)^{1/m}+1$.
• Theorem 1.3: $2^r$ is the exact interval threshold for forcing a monochromatic equation of the displayed form for some $m$.
• Remark 2.2: $r(C_{2\ell+1};q)\leq(4\ell-2)^q(q!)^{1/\ell}+1$, a direct numerator bound relevant to Problem 554 but not a proof of its limiting ratio.