Complete Resolution of the Butler-Costello-Graham Conjecture on Monochromatic Constellations
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917456165994496 |
|---|---|
| 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 |