Hyperplane cuts of an n-cube

Abstract

We say that a hyperplane P of dimension n−1 cuts a hypercube H of dimension n if P does not pass through any of the vertices of H but only through its edges. Two problems are considered: (1) What is the maximal number k of edges of H that P may cut? (2) What is the minimal number m of hyperplanes which cut all the edges of H? The first problem is solved using Baker's generalization of Sperner's lemma. This enables us to give a lower bound on the number m of the second problem.