7.1 Clique Number of Random Graphs

(Throughout this section, log denotes base-2 logarithm.) We now sketch a proof of this theorem. Proof: Let Xk be the number of k-cliques in a graph G drawn from Gn, 2 . By linearity of expectation we have g(k) := E[Xk] = ( n k ) 2−( k 2) and let us define k0(n) to be the largest value of k such that g(k) ≥ 1. An easy calculation (Exercise!) shows that k0(n) ∼ 2 log n. (To see this is plausible, note that for k n, g(k) is roughly n k k! 2 −k2/2 = 2 logn−k 2/2−k log .) Now let c be some small integer constant (independent of n) to be decided later. In order to prove our claim, it is enough to show the following, for some constant c: