Two notions of unit distance graphs

A faithful (unit) distance graph in R d 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 R d is any subgraph of such a graph. 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 R d on n labelled vertices is 2 ( 1 + o ( 1 ) ) d n log 2 n , and give a short proof of the known fact that the number of distance graphs in R d on n labelled vertices is 2 ( 1 − 1 / ⌊ d / 2 ⌋ + o ( 1 ) ) n 2 / 2 . We also study the behavior of several Ramsey-type quantities involving these graphs. 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 R d .