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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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| _version_ | 1866910687213649920 |
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| author | Myzelev, Eric |
| author_facet | Myzelev, Eric |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization of Colorings Obtained by a Method of Szlam Myzelev, Eric Combinatorics 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. |
| title | Characterization of Colorings Obtained by a Method of Szlam |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.04346 |