𝑛-clusters for 1<𝑛<7

Abstract

An n-cluster is a set of n lattice points in R 2 {{\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 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 R m {{\mathbf {R}}^m} .