{"paper":{"title":"Monochromatic unit equilateral triangle on low-dimensional spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"There exists a 2-coloring of the 2-sphere of radius 1/√2 with no monochromatic unit equilateral triangle, but every 2-coloring of the 3-sphere contains one.","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Gennian Ge, Xiaochen Zhao","submitted_at":"2026-05-16T12:21:48Z","abstract_excerpt":"A result of Matou\\v{s}ek and R\\\"odl in 1995 states that for every $\\varepsilon>0$ and every triangle $T$ with circumradius $\\rho(T)$, there exists a dimension $n=n(\\varepsilon,T)$ such that every $2$-coloring of the $n$-dimensional sphere of radius $\\rho(T)+\\varepsilon$, namely $\\mathbb{S}^{n}(\\rho(T)+\\varepsilon)$, contains a monochromatic congruent copy of $T$. In this paper, we determine the exact threshold dimension for the unit equilateral triangle on the sphere $\\mathbb{S}^{n}(1/\\sqrt{2})$: there exists a $2$-coloring of $\\mathbb{S}^{2}(1/\\sqrt{2})$ with no monochromatic unit equilateral"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"there exists a 2-coloring of S^2(1/√2) with no monochromatic unit equilateral triangle, whereas every 2-coloring of S^3(1/√2) contains one.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The unit equilateral triangle is defined via Euclidean chord distances in the ambient R^{n+1}, and the sphere radius 1/√2 is chosen so that the target distance 1 corresponds to a geometrically natural configuration on that sphere.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The exact threshold dimension for monochromatic unit equilateral triangles in 2-colored spheres of radius 1/√2 is 3.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"There exists a 2-coloring of the 2-sphere of radius 1/√2 with no monochromatic unit equilateral triangle, but every 2-coloring of the 3-sphere contains one.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e03df4fd8d436279992c39234061b066b1d64a9e2019da0dc656261a28f288a2"},"source":{"id":"2605.16958","kind":"arxiv","version":1},"verdict":{"id":"26a4464a-c83b-4095-9740-fb1dfbc1b53a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T20:15:58.479495Z","strongest_claim":"there exists a 2-coloring of S^2(1/√2) with no monochromatic unit equilateral triangle, whereas every 2-coloring of S^3(1/√2) contains one.","one_line_summary":"The exact threshold dimension for monochromatic unit equilateral triangles in 2-colored spheres of radius 1/√2 is 3.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The unit equilateral triangle is defined via Euclidean chord distances in the ambient R^{n+1}, and the sphere radius 1/√2 is chosen so that the target distance 1 corresponds to a geometrically natural configuration on that sphere.","pith_extraction_headline":"There exists a 2-coloring of the 2-sphere of radius 1/√2 with no monochromatic unit equilateral triangle, but every 2-coloring of the 3-sphere contains one."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16958/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T20:31:34.764895Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T20:31:19.068805Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T19:51:58.454093Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.232805Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.317531Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"f45629e0faf1436d0d91243ab2997a12ff3e3b3fefef66de172bc15cb6cf787a"},"references":{"count":16,"sample":[{"doi":"","year":2024,"title":"D. Cherkashin and V. Voronov. On the chromatic number of 2-dimensional spheres.Discrete Comput. Geom., 71(2):467–479, 2024","work_id":"111008f9-0967-4f26-b24c-17a880f7b17b","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"G. Currier, K. Moore, and C. H. Yip. Any two-coloring of the plane contains monochromatic 3-term arithmetic progressions.Combinatorica, 44(6):1367–1380, 2024","work_id":"266cbebd-90c6-4b29-a50b-29f2a38f2f8f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1973,"title":"P. Erd˝ os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus. Euclidean Ramsey theorems. I.J. Combinatorial Theory Ser. A, 14:341–363, 1973","work_id":"9d4da6b2-338f-4bbc-8090-ecd5ec75c75e","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1975,"title":"P. Erd˝ os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus. Euclidean Ramsey theorems. II. In A. 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