On existence of integral point sets and their diameter bounds

Abstract

A point set $M$ in $m$-dimensional Euclidean space is called an integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on an $(m-1)$-dimensional hyperplane. We improve the linear lower bound for diameter of planar integral point sets. This improvement takes into account some results related to the Point Packing in a Square problem. Then for arbitrary integers $m \geq 2$, $n \geq m+1$, $d \geq 1$ we give a construction of an integral point set $M$ of $n$ points in $m$-dimensional Euclidean space, where $M$ contains points $M_1$ and $M_2$ such that distance between $M_1$ and $M_2$ is exactly $d$.