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.
翻译:硬向日葵是组合式日葵花的概括化,它应用在单线电路复杂度、 DNF 蒸发、随机抽取器以及Erd\H{{o}s-Rado日葵花预测的最新进展。最近Alweiss、Lovet、Wu和张的突破使Alweiss、Lovet、Wu和Zhang在排除强向日葵的美元定价系统的最大尺寸上得到了改进。在本文中,我们利用这一结果获得一个在单线电路复杂度、DNNNNF 蒸汽、随机抽取器提取器以及Erd\h{H{{{%1/3-o(1)}等功能上较低的单线电路尺寸,从而改进了对安德烈夫、哈尼克和拉兹的已知的美元最大值。我们还展示了在相关聚向型光线的单线算电路程大小上较低的美元值。最后,我们引入了一种坚固的圆向向花的理念,并用它来证明“$-O+3”的硬值。