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Optimization Tools for Computing Colorings of $[1,\cdots ,n]$ with Few Monochromatic Solutions on $3$-variable Linear Equations

Jesús A. De Loera, Denae Ventura, Liuyue Wang, William J. Wesley

Abstract

A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $E$ there is a threshold value $R_k(E)$ (the Rado number of $E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \ge R_k(E)$, there exists at least one monochromatic solution. But one can further ask, how many monochromatic solutions is the minimum possible in terms of $n$? Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.

Optimization Tools for Computing Colorings of $[1,\cdots ,n]$ with Few Monochromatic Solutions on $3$-variable Linear Equations

Abstract

A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations there is a threshold value (the Rado number of ) such that for any -coloring of the integers in the interval , with , there exists at least one monochromatic solution. But one can further ask, how many monochromatic solutions is the minimum possible in terms of ? Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.

Paper Structure

This paper contains 7 sections, 10 theorems, 66 equations, 5 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Upper Bounds on $M_{\cal E}(n)$ for 3-term linear equations
  3. Max-Cut and Best Colorings via Integer Optimization
  4. Optimal Colorings with a Semidefinite Program Trace Bound Method
  5. More than two colors in Schur's equation
  6. Data for SDP lower bounds
  7. Integer Programming Results

Key Result

Theorem 1

Let $a$ be a positive integer and ${\cal E}$ be the equation $ax+ay=z$. The following holds but the bound is not tight.

Figures (5)

  • Figure 1: Coloring $\chi$ used in Theorems \ref{['upperbax+ay']} and \ref{['upperbax-ay']} where all multiples of $a$ are red and the rest is blue.
  • Figure 2: Coloring $\chi_0$, where $a$ is odd, and integers that are $1,3,5,\dots,a-2 \pmod a$ or $a \pmod{2a}$ are red, and integers that are $2,4,6\dots,a-1 \pmod a$ or $0 \pmod{2a}$ are blue.
  • Figure 3: Coloring $\chi$, where the interval $[1,s] \cup [t,n]$ is red and $(s,t)$ blue, where $s = \frac{ 4n(a + 1)}{a^2(4 + a)}, \quad t = \frac{2n}{a}.$
  • Figure 4: Coloring $\chi$ used in Theorem \ref{['upperbax+by']} where $[1, \lfloor\frac{n}{a(a+b)}\rfloor]$ and $[\lfloor \frac{n}{a}\rfloor +1,n]$ are red blocks and the rest is blue.
  • Figure 5: The weighted graph $G_{x+y=z,5}$.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • proof : Proof of Theorem \ref{['upperbax+ay']}
  • proof : Proof of Theorem \ref{['upperbax-ay']}
  • Lemma 8
  • ...and 11 more