Work

The Erdős Unit Distance Problem for Small Point Sets

Year: 2026

Type: article

Abstract: 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.

Source: Studia Scientiarum Mathematicarum Hungarica

DOI: 10.1556/012.2026.04350

Authors: Boris Alexeev, Dustin G. Mixon, Hans Parshall

Institution: The Ohio State University

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Air Force Office of Scientific Research

Awards: G1523888516, ATD: Collaborative Research: Theory and Algorithms for Real-Time Threat Detection from Massive Data Streams, G760981969, Collaborative Research: ATD: Principled machine learning and packing subspaces for improved spatiotemporal data processing