The Erdős unit distance problem asks to determine the maximum number 
of occurrences of the same distance among 
points in the plane. It is equivalent to finding a maximally dense unit-distance graph on 
vertices.
Erdős (1946) proved a lower bound of 
and he offered a $500 prize for a matching upper bound. The best upper bound current known is 
(Szemerédi 2016).
Exact values of 
for 
were obtained by Schade (1993) together with the complete sets of densest graphs for 
. Agoston and Pálvőlgyi (2022) subsequently determined the exact value of 
, and Alexeev et al. (2025) found 
for 
as well as a complete set of densest graphs. All but one of these graphs (one of the 7 on 17 vertices whose unit-distance embedding does not reside within the Moser ring) were previously discovered by Engel et al. (2024).
For 
, 2, ..., 
is given by 0, 1, 3, 5, 7, 9, 12, 14, 18, 20, 23, 27, 30, 33, 37, 41, 43, 46, 50, 54, 57, ... (OEIS A186705). The corresponding numbers of nonisomorphic maximally dense unit-distance graphs are 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 7, 16, 3, 1, 5, ... (OEIS A385657; Alexeev et al. 2025).
Points." Amer. Math. Monthly 53, 248-250, 1946.Schade, C. "Exakte Maximale Anzahlen Gleicher Abstände." Thesis. Braunschweig, Germany: TU Braunschweig, 1993.Sloane, N. J. A. Sequences