  Robust sunflowers are a generalization of combinatorial sunflowers that have
applications in monotone circuit complexity, DNF sparsification, randomness
extractors, and recent advances on the Erd\H{o}s-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 $\exp(\Omega(n))$ 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 $n^{\Omega(k)}$ lower bound on the
monotone circuit size of the CLIQUE function for all $k \le n^{1/3-o(1)}$,
strengthening the bound of Alon and Boppana.
