Two-point concentration in random geometric graphs

A random geometric graph G n is constructed by taking vertices X 1,…,X n ¿R d at random (i.i.d. according to some probability distribution ¿ with a bounded density function) and including an edge between X i and X j if ¿X i -X j ¿ <r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r=r(n) is chosen such that nr d = o(lnn) then the probability distribution of the clique number ¿(G n ) becomes concentrated on two consecutive integers and we show that the same holds for a number of other graph parameters including the chromatic number ¿(G n ).