Diameter-Ramsey triangles below the $135^\circ$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913058297741312 |
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| author | Mao, Yaping |
| author_facet | Mao, Yaping |
| contents | A finite Euclidean set is diameter-Ramsey if, for every number of colors, some finite same-diameter witness has the property that every coloring of the witness contains a monochromatic congruent copy of the set. Frankl, Pach, Reiher and Rödl asked whether any obtuse triangle is diameter-Ramsey. We prove the stronger statement that every non-degenerate triangle whose largest angle is strictly smaller than $135^\circ$ is diameter-Ramsey. Together with the theorem of Corsten and Frankl that triangles with an angle larger than $135^\circ$ are not diameter-Ramsey, this gives the sharp classification for the two open angular ranges on either side of $135^\circ$. The proof uses a weighted $k$-subset configuration with non-negative coefficients; a finite binary-tree construction realizes the required two prescribed overlaps, and the ordinary hypergraph Ramsey theorem then forces a monochromatic copy of the triangle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22090 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Diameter-Ramsey triangles below the $135^\circ$ Mao, Yaping Combinatorics A finite Euclidean set is diameter-Ramsey if, for every number of colors, some finite same-diameter witness has the property that every coloring of the witness contains a monochromatic congruent copy of the set. Frankl, Pach, Reiher and Rödl asked whether any obtuse triangle is diameter-Ramsey. We prove the stronger statement that every non-degenerate triangle whose largest angle is strictly smaller than $135^\circ$ is diameter-Ramsey. Together with the theorem of Corsten and Frankl that triangles with an angle larger than $135^\circ$ are not diameter-Ramsey, this gives the sharp classification for the two open angular ranges on either side of $135^\circ$. The proof uses a weighted $k$-subset configuration with non-negative coefficients; a finite binary-tree construction realizes the required two prescribed overlaps, and the ordinary hypergraph Ramsey theorem then forces a monochromatic copy of the triangle. |
| title | Diameter-Ramsey triangles below the $135^\circ$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.22090 |