The Erdős Unit Distance Problem for Small Point Sets

We improve the best known upper bound on the number of edges in a unit-distance graph on n vertices for each n ∈ {16, … , 30}. When n ≤ 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 principal technique is to more efficiently generate ℱ -free graphs for a set of forbidden subgraphs 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.