Monotone Circuit Lower Bounds from Robust Sunflowers
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Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity, DNF sparsification, randomness extractors, and recent advances on the Erdős-Rado sunflower conjecture. The recent breakthrough of Alweiss, Lovett, Wu and Zhang 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 and Harnik and Raz. We also show an exp(Ω(n)) lower bound on the monotone arithmetic circuit size of a related polynomial. Finally, we introduce a notion of robust clique-sunflowers and use this to prove an n^Ω(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.
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