The multiplicity of the two smallest distances among points (Q1297473)

The author studies distances occuring in configurations of \(n\) points in the plane, where the minimal distance between two points is assumed to be \(1=:d_1\). The multiplicity of the smallest occuring distances and \(d_1\) and \(d_2\) is smaller than \(6n\) and maximized by the triangular lattice with \(d_2= \sqrt 3\). Partially answering a question of P. Erdős and K. Vesztergombi, the author proves that \(d_2\neq\sqrt 3\) implies that the multiplicity \(d_1\) and \(d_2\) is at most \(4n\) unless \(d_2\) is \((\sqrt 5+1)/2\) or \(1/(2\sin 15)\). In the first case, the multiplicity is at most \(4,5 n\). In addition, the author presents same interesting extremal point configurations for different values of \(d_2\).