Multiplicities of interpoint distances in finite planar sets (Q1894357)

For a set \(X\) of \(n\) points in the plane, let \(d_1, \ldots, d_m\) denote the different positive distances between the points of \(X\), and \(r_k\) the multiplicity of \(d_k\). The authors study the vector \(r(X) = (r_1, \ldots, r_m)\), where the numbering is chosen such that \(r_1 \geq r_2 \geq \cdots \geq r_m\). The case where \(X\) is the set \(V\) of vertices of a convex polygon is considered particularly. For \(n = 5\) and \(m \in \{2,3\}\), the possible vectors \(r(X)\) and \(r(V)\) are completely specified. For \(n = 6\), it is shown that \(r(X)\) cannot be equal to (7,7,1). There is a discussion of some known results and several challenging conjectures which are related to this topic.