The Monotone Complexity of k-clique on Random Graphs

It is widely suspected that Erdös-Renyi random graphs are a source of hard instances for clique problems. Giving further evidence for this belief, we prove the first average-case hardness result for the k-clique problem on monotone circuits. Specifically, we show that no monotone circuit of size O(n^k/4) solves the k-clique problem with high probability on G(n,p) for two sufficiently far-apart threshold functions p(n) (for instance n^-2/(k-1) and 2n^-2/(k-1)). Moreover, the exponent k/4 in this result is tight up to an additive constant. One technical contribution of this paper is the introduction of quasi-sunflowers, a new relaxation of sunflowers in which petals may overlap slightly on average. A "quasi-sunflower lemma" (à la the Erdös-Rado sunflower lemma) leads to our novel lower bounds within Razborov\´s method of approximations.