Colouring proximity graphs in the plane

Given a set V of points in the plane and given d>0, let G(V, d) denote the graph with vertex set V and with distinct vertices adjacent whenever the Euclidean distance between them is less than d. We are interested in colouring such 'proximity' graphs. One application where this problem arises is in the design of cellular telephone networks, where we need to assign radio channels (colours) to transmitters (points in V) to avoid interference. We investigate the case when the set V has finite positive upper density σ, and d is large. We find that, as d → ∞, the chromatic number χ divided by d2 tends to the limit σ(√3)/2, and the ratio of the chromatic number χ to the clique number ω tends to 2(√3)/π ~ 1.103.