The Erdős unit distance problem asks to determine the maximum number u(n) of occurrences of the same distance among n points in the plane. It is equivalent to finding a maximally dense unit-distance graph on n vertices. Erdős proved a lower bound of u(n) = n^(1 + Ω(1/ln ln n)) and he offered a $500 prize for a matching upper bound. The best upper bound current known is u(n) = O(n^(4/3)).
