The Erdős unit distance problem for small point sets

Abstract

We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in \{16,\dots ,30\}$. When $n\le 21$, 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.

18 pages, 3 tables, 63 figures; v2 has only minor changes; see also ancillary file graph6.txt