DP-Colorings of Uniform Hypergraphs and Splittings of Boolean Hypercube into Faces

We develop a connection between DP-colorings of k-uniform hypergraphs of order n and coverings of n-dimensional Boolean hypercube by pairs of antipodal (n-k)-dimensional faces. Bernshteyn and Kostochka established a lower bound on the number of edges in a non-2-DP-colorable k-uniform hypergraph namely, 2(k-1) for odd k and 2(k-1) + 1 for even k. They proved that these bounds are tight for k = 3, 4. In this paper, we prove that the bound is achieved for all odd k >= 3.