The chromatic numbers of some Euclidean distance graphs (Q2719864)

For \((X,\rho)\) a metric space and \(s>0\), let \((X,s)\) be the graph whose vertices are the points of \(X\), with \(x\) and \(y\) adjacent precisely when \(\rho(x,y)= s\). In this paper, \(X\) is usually a subset of some Euclidean space \(\mathbb{R}^n\), with \(\rho\) the usual Euclidean metric. The authors study the chromatic number of \((X,s)\). They present some results, particularly for \(n= 2\), and they state six open problems.