Székely distinct-distance bound (${\max }_{a\in P}|\{|a-b|:b\in P\setminus \{a\}\}|\ge c|P{|}^{4/5}$)

For a finite set of at least two points in the plane, some point determines the displayed number of different positive Euclidean distances. The circular-arc graph for distinct distances has quadratically many edges. The rich-bisector deletion bound removes high-multiplicity edges while retaining that order of growth. The crossing lemma for multigraphs then compares its lower crossing bound with the quadratic count of pairs of circles. This strengthens the corresponding bound for the whole distinct-distance set. The original result is Theorem11 of Székely's 1997 paper.