$n$-clusters for $1<n<7$

Abstract:

An n-cluster is a set of n lattice points in ${{\mathbf {R}}^2}$ with no 3 points collinear, no 4 points concyclic, and where all mutual distances are nonzero integers. The initial discoveries of n-clusters for $1 < n < 7$ are reported. A progress report on small 6-clusters and the search for a 7-cluster is described. A question on the impact of the lattice point restriction is raised, and the definition of n-cluster is generalized to ${{\mathbf {R}}^m}$.