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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.04346 |
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Table of Contents:
- 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.