More distinct distances under local conditions
The paper is related to the Erdős problem of distinct distances. In 1946 Erdős asked to estimate the minimum number of distinct distances determined by a set of \(n\) points on the plane. For integers \(p\) and \(q\) with \(q \leq \binom{p}2\), let \(D(n,p,q)\) denote the minimum number of distinct distances determined by a set \(V\) of \(n\) points on the plane with the property that any \(p\) points from \(V\) determine at least \(q\) distinct distances. Some lower and upper bounds have been obtained for \(D(n,p,q)\) under certain local conditions by different authors. In this paper, the following result is proved. Let \(V\) be a set of \(n\) points on the plane such that any \(p\) points induce at least \(\binom{p}2 - p + 6\) distinct distances. Then \(V\) determines at least \(n^{\frac87 - o(1)}\) distinct distances as \(n\) tends to infinity. In the proof, some graph-theoretic tools are used.
scientific article; zbMATH DE number 6909501