Sets in the Plane with Many Concyclic Subsets

Abstract

We study sets of points in the Euclidean plane having property $R(t,s)$: every $t$-tuple of its points contains a concyclic $s$-tuple. Typical examples of the kind of theorems we prove are: a set with $R(19,10)$ must have all its points on two circles or all its points, with the exception of at most 9, are on one circle; of a set with $R(8,5)$ and $N\geq 28$ points at least $N-3$ points lie on one circle; a set of at least 109 points with $R(7,4)$ has $R(109,7)$. We added some results on the analogous configurations in 3-space.