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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.02115 |
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| _version_ | 1866917456165994496 |
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| author | Yang, Gang Mao, Yaping |
| author_facet | Yang, Gang Mao, Yaping |
| contents | A constellation pattern is a finite increasing rational sequence \(Q=[0=q_0<q_1<\cdots<q_k=1]\), and a \(Q\)-constellation in \([n]\) is obtained by scaling and translating a rational pattern $Q$, with key examples including arithmetic progressions. In 2010, Butler, Costello, and Graham proposed a conjecture, that is, for any constellation pattern $Q$ there is a coloring pattern of $[n]$ that has $γn^2+o\left(n^2\right)$ monochromatic constellations, where $γ$ is smaller than the coefficient for a random coloring. In this paper, we confirm this conjecture. As applications of this conjecture, we obtain interval-uncommon translation-invariant linear systems associated with rational constellations and a ground-state bound for deterministic arithmetic hypergraph spin systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02115 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete Resolution of the Butler-Costello-Graham Conjecture on Monochromatic Constellations Yang, Gang Mao, Yaping Combinatorics A constellation pattern is a finite increasing rational sequence \(Q=[0=q_0<q_1<\cdots<q_k=1]\), and a \(Q\)-constellation in \([n]\) is obtained by scaling and translating a rational pattern $Q$, with key examples including arithmetic progressions. In 2010, Butler, Costello, and Graham proposed a conjecture, that is, for any constellation pattern $Q$ there is a coloring pattern of $[n]$ that has $γn^2+o\left(n^2\right)$ monochromatic constellations, where $γ$ is smaller than the coefficient for a random coloring. In this paper, we confirm this conjecture. As applications of this conjecture, we obtain interval-uncommon translation-invariant linear systems associated with rational constellations and a ground-state bound for deterministic arithmetic hypergraph spin systems. |
| title | Complete Resolution of the Butler-Costello-Graham Conjecture on Monochromatic Constellations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.02115 |