# DP-colorings of uniform hypergraphs and splittings of Boolean hypercube into faces

## TL;DR

This paper explores the relationship between DP-colorings of uniform hypergraphs and hypercube face coverings, establishing tight bounds for all odd uniformities and connecting combinatorial hypergraph properties with geometric hypercube structures.

## Contribution

It extends previous bounds on non-2-DP-colorable hypergraphs, proving the bounds are tight for all odd uniformities, and links hypergraph colorings to hypercube face coverings.

## Key findings

- Bound on edges is achieved for all odd k ≥ 3.
- Established a connection between hypergraph DP-colorings and hypercube face coverings.
- Proved tight bounds for non-2-DP-colorable hypergraphs for all odd k.

## Abstract

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 that the lower bound on edges in a non-2-DP-colorable $k$-uniform hypergraph is equal to $2^{k-1}$ 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 $k\geq 3$.