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Characterization of Colorings Obtained by a Method of Szlam

Eric Myzelev

Abstract

Szlam's Lemma began life as a way of getting upper bounds on the chromatic numbers of distance graphs in normed vector spaces. Now analogs are available in a variety of hypergraph settings, but the method always involves a shrewdly chosen 2-coloring of the vertex set of a hypergraph, together with a subset of the vertex set which satisfies certain requirements with reference to the 2-coloring. From these ingredients a proper coloring of the hypergraphs is cooked up. In this paper, we separate the process from the conclusion of Szlam's Lemma by defining Szlam colorings of the vector spaces $\mathbb R^d$, and then a more regimented variety of these, which we call ordered Szlam colorings, which we characterize.

Characterization of Colorings Obtained by a Method of Szlam

Abstract

Szlam's Lemma began life as a way of getting upper bounds on the chromatic numbers of distance graphs in normed vector spaces. Now analogs are available in a variety of hypergraph settings, but the method always involves a shrewdly chosen 2-coloring of the vertex set of a hypergraph, together with a subset of the vertex set which satisfies certain requirements with reference to the 2-coloring. From these ingredients a proper coloring of the hypergraphs is cooked up. In this paper, we separate the process from the conclusion of Szlam's Lemma by defining Szlam colorings of the vector spaces , and then a more regimented variety of these, which we call ordered Szlam colorings, which we characterize.

Paper Structure

This paper contains 3 sections, 4 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. The Question
  3. The Answer

Key Result

Theorem 1.1

Suppose that $\mathbb{R}^2$ is colored with red and blue so that the Euclidean distance 1 is forbidden for blue. Then the red set contains a translate of each 3-point set in $\mathbb{R}^2$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3: Szlam's Lemma
  • proof
  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • proof
  • Example 3.2
  • Remark 3.2.1
  • ...and 1 more