Sets in the plane with many concyclic subsets (Q2571284)

A set \(S\) of points in the Euclidean plane is said to have the property \(R(t,s)\) if every \(t\)-subset of \(S\) has a concyclic \(s\)-subset. Here, a \(k\)-set stands for a set with \(k\) elements. It is clear that if \(t \geq s\) and \(s \leq 3\), then every set has the property \(R(t,s)\). The paper under review investigates the consequences of \(R(t,s)\) for several values of \(t\) and \(s\). Here are typical examples of the results: (i) If a set has the property \(R(19,10)\), then either its points lie on two circles or all except at most 9 points lie on a circle. (ii) If an \(N\)-set has the property \(R(8,5)\) and if \(N \geq 28\), then it has the property \(R(N,N-3)\). (iii) If an \(N\)-set has the property \(R(7,4)\) and if \(N \geq 109\), then it has the property \(R(109,7)\). The last section is devoted to the analogous issues in the 3-space. MAPLE is used at times to carry out tedious calculations.