Clique Is Hard on Average for Regular Resolution

We prove that for $k\ll \sqrt[4]{n}$ regular resolution requires length $n^{\mathrm{\Omega }(k)}$ to establish that an Erd\H{o}s-R'enyi 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 $n^{\mathrm{\Omega }(k)}$ lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.