Two-point concentration in random geometric graphs

A random geometric graph Gn is constructed by taking vertices X1, . . . , Xn ∈ R at random (i.i.d. according to some probability distribution ν with a bounded density function) and including an edge between Xi and Xj if ‖Xi−Xj‖ < r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r = r(n) is chosen such that nr = o(lnn) then the probability distribution of the clique number ω(Gn) becomes concentrated on two consecutive integers and we show that the same holds for a number of other graph parameters including the chromatic number χ(Gn).