We prove that for k ≪ ⁴ n regular resolution requires length n^Ω(k) to establish that an Erdős–Rényi graph with appropriately chosen edge density does not contain a k-clique. This lower bound is optimal up to the multiplicative constant in the exponent, and also implies unconditional n^Ω(k) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.