A lower bound for essential covers of the cube

05/28/2021

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by   Gal Yehuda, et al.

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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 Ω(n^0.5). Later on, this notion found several applications in complexity theory. We improve the lower bound to Ω(n^0.52), and describe two applications.