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

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.