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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.15147","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T17:49:42Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"fa0e689de7314aaa9a400c21b9b81c228aa37e5e3372ba8b61ed8dc054384368","abstract_canon_sha256":"7ddfed18f5a04c8e9c3d5f20bac5da50f42db08844541595b464a2d907f4d450"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.2","canonical_sha256":"a748d66d88a895030ff3b483092eb39f42f1eb297a68ef42cb433040ca2a7f6f","last_reissued_at":"2026-05-17T21:57:18.826133Z","signature_status":"unsigned_v0","first_computed_at":"2026-05-17T21:40:25.506232Z"},"graph_snapshot":{"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. 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