Two notions of unit distance graphs

A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particular, we show that for any fixed d the number of faithful distance graphs in Rd on n labelled vertices is 2(1+o(1))dn log2 n, and give a short proof of the known fact that the number of distance graphs in Rd on n labelled vertices is 2(1−1/bd/2c+o(1))n2/2. We also study the behavior of several Ramsey-type quantities involving these graphs. In the second part of the paper we discuss the problem of determining the minimum possible number of edges of a graph which is not isomorphic to a faithful distance graph in Rd.