Distinct distances in planar point sets with forbidden 4-point patterns

Abstract

A point set in a metric space is said to have property Φ(4,5) if every 4 elements determine at least 5 distinct distances. According to an old conjecture of Erdős (1986 or earlier), a set of n points in the Euclidean plane satisfying this restriction determines Ω(n2) distinct distances. This property (restriction) is shown to be equivalent to forbidding eight 4-element patterns, πi, i=1,…,8 (described in Section 2, Lemma 1). The existence of n-element point sets without the three patterns π1,π2,π3, that determine only o(n2) distinct distances was previously known. Here we exhibit n-element point sets without the seven patterns π1,π3,π4,π5,π6,π7,π8, that determine only o(n2) distinct distances. The existence of point sets missing all eight forbidden patterns and determining only o(n2) distinct distances remains open.