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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.","authors_text":"Cosmin Pohoata, Eion Mulrenin, Michael Zheng, Rafael Miyazaki","cross_cats":["math.NT"],"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.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T17:49:42Z","title":"Improved Ramsey bounds for generalized Schur equations"},"references":{"count":22,"internal_anchors":0,"resolved_work":22,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"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","year":1972},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"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","year":2022},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"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","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"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","year":2020},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"R. B. Crittenden and C. L. Vanden Eynden,A proof of a conjecture of Erdős, Bull. 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