On the Probability of Independent Sets in Random Graphs

Let k be the asymptotic value of the independence number of the random graph G(n, p). We prove that if the edge probability p(n) satisfies p(n) ≫ n^-2/5ln^6/5n then the probability that G(n, p) does not contain an independent set of size k - c, for some absolute constant c > 0, is at most exp{ -cn²/(k⁴p)}. We also show that the obtained exponent is tight up to logarithmic factors, and apply our result to obtain new bounds on the choice number of random graphs. We also discuss a general setting where our approach can be applied to provide an exponential bound on the probability of certain events in product probability spaces.