Essential covers were introduced by Linial and Radhakrishnan as a model that captures two complementary properties: (1) all variables must be included and (2) no element is redundant. In their seminal paper, they proved that every essential cover of the $n$-dimensional hypercube must be of size at least $\Omega(n^{0.5})$. Later on, this notion found several applications in complexity theory. We improve the lower bound to $\Omega(n^{0.52})$, and describe two applications.

中文翻译：

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立方体基本覆盖的下限

基本覆盖由 Linial 和 Radhakrishnan 引入，作为捕获两个互补属性的模型：(1) 必须包含所有变量，(2) 没有元素是多余的。在他们开创性的论文中，他们证明了 $n$ 维超立方体的每个基本覆盖的大小必须至少为 $\Omega(n^{0.5})$。后来，这个概念在复杂性理论中得到了一些应用。我们将下界改进为 $\Omega(n^{0.52})$，并描述了两个应用。