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

Gang Yang, Yaping Mao

Abstract

A constellation is a subset of $[n]=\{1,2, \ldots, n\}$ formed by scaling and translating a rational pattern $Q=\left[0, q_1, \ldots, q_{k-1}, 1\right]$, 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.

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

Abstract

A constellation is a subset of formed by scaling and translating a rational pattern , with key examples including arithmetic progressions. In 2010, Butler, Costello, and Graham proposed a conjecture, that is, for any constellation pattern there is a coloring pattern of that has monochromatic constellations, where is smaller than the coefficient for a random coloring. In this paper, we confirm this conjecture.

Paper Structure

This paper contains 13 sections, 13 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ordered and actual copies
  4. Total counts and the random main term
  5. A continuous model and the discrete-to-continuous transfer
  6. Uniform residue-class Riemann sums
  7. Two-dimensional discrete sums versus the continuous functional
  8. Second variation at the random point
  9. An explicit negative direction
  10. The first mixed family
  11. The second mixed family
  12. A negative direction
  13. From a negative direction to deterministic colorings

Key Result

Theorem 1.1

Let $Q=[q_0=0<q_1<\cdots<q_k=1]$, where $q_i\in\mathbb{Q}$, with $k\ge 2$. Then there exist a constant $\delta_{Q}>0$ and a sequence of two-colorings $\chi_n:[n]\to\{\pm1\}$ such that where and $D$ is the least common denominator of the $q_i$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (29)

  • Definition 1
  • Conjecture 1: Butler-Costello-Graham Conjecture, BCG10
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Lemma 3.1
  • ...and 19 more