= Székely distinct-distance bound
{c}
{title2=$\max_{a\in P}|\{|a-b|:b\in P\setminus\{a\}\}|\geq c|P|^{4/5}$}
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 https://www.cs.tau.ac.il/~michas/szekely.pdf[Székely's 1997 paper].
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