Monotone Circuit Lower Bounds from Robust Sunflowers

Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity Rossman (SIAM J. Comput. 43:256-279, 2014), DNF sparsification Gopalan et al. (Comput. Complex. 22:275-310 2013), randomness extractors Li et al. (In: APPROX-RANDOM, LIPIcs 116:51:1-13, 2018), and recent advances on the Erdos-Rado sunflower conjecture Alweiss et al. (In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC. Association for Computing Machinery, New York, NY, USA, 2020) Lovett et al. (From dnf compression to sunflower theorems via regularity, 2019) Rao (Discrete Anal. 8,2020). The recent breakthrough of Alweiss, Lovett, Wu and Zhang Alweiss et al. (In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC. Association for Computing Machinery, New York, NY, USA, 2020) gives an improved bound on the maximum size of a w-set system that excludes a robust sunflower. In this paper, we use this result to obtain an exp(n(1/2-o(1))) lower bound on the monotone circuit size of an explicit n-variate monotone function, improving the previous best known exp(n(1/3-o(1))) due to Andreev (Algebra and Logic, 26:1-18, 1987) and Harnik and Raz (In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, ACM, New York, 2000). We also show an exp(Omega(n)) lower bound on the monotone arithmetic circuit size of a related polynomial via a very simple proof. Finally, we introduce a notion of robust clique-sunflowers and use this to prove an n(Omega(k)) lower bound on the monotone circuit size of the CLIQUE function for all k <= n(1/3-o(1)), strengthening the bound of Alon and Boppana (Combinatorica, 7:1-22, 1987).