Complete Resolution of the Butler-Costello-Graham Conjecture on Monochromatic Constellations

TL;DR

This paper proves the Butler-Costello-Graham Conjecture, showing that for any rational constellation pattern, there exists a coloring with fewer monochromatic constellations than random coloring, with applications in linear systems and hypergraph spin systems.

Contribution

The paper confirms the longstanding conjecture, establishing the existence of colorings with fewer monochromatic constellations for any rational pattern, advancing combinatorial and hypergraph theory.

Findings

Confirmed the Butler-Costello-Graham Conjecture.

Constructed colorings with fewer monochromatic constellations than random.

Derived applications in linear systems and hypergraph spin models.

Abstract

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 $\gamma n^2+o\left(n^2\right)$ monochromatic constellations, where $\gamma$ 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.