The Erd\H{o}s unit distance problem for small point sets

TL;DR

This paper advances the understanding of the Erdős unit distance problem for small point sets by improving upper bounds, fully enumerating densest graphs for certain sizes, and developing efficient computational methods for graph analysis.

Contribution

It introduces improved bounds and enumeration for small point sets and develops a more efficient computational approach for determining unit-distance graphs.

Findings

Bounds match known lower bounds for n ≤ 21

Complete enumeration of densest graphs for n ≤ 21

Development of a more efficient graph embedding algorithm

Abstract

We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest unit-distance graphs in these cases. On the combinatorial side, our principle technique is to more efficiently generate $\mathcal{F}$-free graphs for a set of forbidden subgraphs $\mathcal{F}$. On the algebraic side, we are able to determine programmatically whether many graphs are unit-distance, using a custom embedder that is more efficient in practice than tools such as cylindrical algebraic decomposition.