{"paper":{"title":"Improved Ramsey bounds for generalized Schur equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For N larger than (2m+1)^r times (r!)^{1/m}, any r-coloring of [N] forces a monochromatic solution to x1+...+x_{m+1}=y1+...+ym.","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Cosmin Pohoata, Eion Mulrenin, Michael Zheng, Rafael Miyazaki","submitted_at":"2026-05-14T17:49:42Z","abstract_excerpt":"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\n  \\[\n  x_1 + \\dots + \\dots x_{m+1} = y_1 + \\dots + y_m.\n  \\]\n  This generalizes and improves recent results of Ko\\'scuiszko. 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."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for m, r ∈ 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 x1 + ⋯ + x_{m+1} = y1 + ⋯ + ym","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The derivation of the explicit bound from standard combinatorial tools (pigeonhole or iterative coloring arguments) holds without hidden dependencies on m or r that would invalidate the inequality for large values.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Improved explicit upper bounds on the Ramsey numbers guaranteeing monochromatic solutions to x1+...+x_{m+1}=y1+...+ym in r-colorings of [N], with the bound N>(2m+1)^r (r!)^{1/m} and optimality of N=2^r for some m.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For N larger than (2m+1)^r times (r!)^{1/m}, any r-coloring of [N] forces a monochromatic solution to x1+...+x_{m+1}=y1+...+ym.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f94bae23144e8699fa3a4e5d1c8c5d338fddd4604dcf3871254d5af5b2af8650"},"source":{"id":"2605.15147","kind":"arxiv","version":1},"verdict":{"id":"39d650d6-1fcf-4860-bd7d-282106c73f25","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:58:07.160165Z","strongest_claim":"for m, r ∈ 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 x1 + ⋯ + x_{m+1} = y1 + ⋯ + ym","one_line_summary":"Improved explicit upper bounds on the Ramsey numbers guaranteeing monochromatic solutions to x1+...+x_{m+1}=y1+...+ym in r-colorings of [N], with the bound N>(2m+1)^r (r!)^{1/m} and optimality of N=2^r for some m.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The derivation of the explicit bound from standard combinatorial tools (pigeonhole or iterative coloring arguments) holds without hidden dependencies on m or r that would invalidate the inequality for large values.","pith_extraction_headline":"For N larger than (2m+1)^r times (r!)^{1/m}, any r-coloring of [N] forces a monochromatic solution to x1+...+x_{m+1}=y1+...+ym."},"references":{"count":22,"sample":[{"doi":"","year":1972,"title":"H.AbbottandD.Hanson,A problem of Schur and its generalizations, ActaArith.20(1972), 175–187; MR0319934 2","work_id":"18c00175-257c-4e62-975c-001ade910fd3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel, J. Tomasik,New lower bounds for Schur and weak Schur numbers(2022), preprint available athttps://arxiv.org/abs/2112.031752","work_id":"6248d51e-eb2e-4a06-83dc-040c54d589c8","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"M. Axenovich, W. Cames von Batenburg, O. Janzer, L. Michel, and M. RundströmAn improved upper bound for the multicolor Ramsey number of odd cycles(2025), preprint available athttps://arxiv.org/abs/251","work_id":"36f468b0-799c-4040-b5d1-ea8c2fe721ab","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe, and M. Tiba,Covering intervals with arithmetic progressions, Acta Math. Hungar.161(2020), 197–200; MR4110365 7","work_id":"8438956e-f144-41f1-af31-8615c662c61a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1969,"title":"R. B. Crittenden and C. L. Vanden Eynden,A proof of a conjecture of Erdős, Bull. Amer. Math. Soc.75(1969), 1326–1329; MR0249351 3, 7","work_id":"8d6f5daa-df44-46b0-8af0-74c487b993cb","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":22,"snapshot_sha256":"fa6b071a96c7ee53e2c185b60ef8ae0504d53cc77635b0b8df9a2d5d3c8f6288","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}