The Erdős unit distance problem for small point sets
The Erdős unit distance problem for small point setsWe improve the best known upper bound on the number of edges in a unit-distance graph on n vertices for each $n\in\{15,\dotsc,30\}$. 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.
Boris Alexeev and Dustin G. Mixon
The Erdős unit distance problem for small point sets
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