Albert Atserias, Ilario Bonacina, Susanna F. de Rezende, Massimo Lauria, Jakob Nordström, A. Razborov
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We prove that for k ≪ n1/4 regular resolution requires length nΩ(k) to establish that an Erdos-Renyi 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 nΩ(k) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.
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