Characterization of Colorings Obtained by a Method of Szlam

TL;DR

This paper introduces and characterizes Szlam colorings and ordered Szlam colorings in vector spaces, extending Szlam's Lemma to hypergraph settings and providing a structured approach to hypergraph coloring.

Contribution

It defines Szlam colorings and ordered Szlam colorings in vector spaces, separating the coloring process from Szlam's Lemma conclusion and characterizing these colorings.

Findings

Introduction of Szlam colorings in  spaces

Definition and characterization of ordered Szlam colorings

Extension of Szlam's Lemma to hypergraph settings

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.