The amount of hyperplanes that are needed in order to cover the Boolean cube has been studied in various contexts. Linial and Radhakrishnan introduced the notion of essential covers. An essential cover is a collection of hyperplanes that form a minimal cover of the vertices of the hypercube, and every coordinate is influential in at least one of the hyperplanes. Linial and Radhakrishnan proved using algebraic tools that every essential cover of the n-cube must be of size at least \(\Omega (\sqrt{n})\). We devise a stronger lower bound method, and show that the size of every essential cover is at least \(\Omega (n^{0.52})\). This result has implications in proof complexity, because essential covers have been used to prove lower bounds for several proof systems.

已经在各种背景下研究了覆盖布尔立方体所需的超平面的数量。 Linial 和 Radhakrishnan 引入了基本封面的概念。基本覆盖是形成超立方体顶点的最小覆盖的超平面的集合，并且每个坐标至少对其中一个超平面有影响。 Linial 和 Radhakrishnan 使用代数工具证明了n立方体的每个基本覆盖层的大小必须至少为 \(\Omega (\sqrt{n})\)。我们设计了一个更强的下界方法，并证明每个基本覆盖的大小至少为 \(\Omega (n^{0.52})\)。这个结果对证明复杂性有影响，因为基本覆盖已被用来证明多个证明系统的下界。