Linear polychromatic colorings of hypercube faces

TL;DR

This paper establishes a new lower bound for the maximum number of colors in polychromatic colorings of hypercube faces, ensuring every embedded lower-dimensional hypercube contains all colors.

Contribution

It introduces a novel lower bound on the polychromatic number for coloring hypercube faces, advancing understanding of hypercube face colorings.

Findings

New lower bound on p^l(d) for l > 1

Enhanced understanding of face colorings in hypercubes

Implications for combinatorial hypercube coloring problems

Abstract

A coloring of the $\ell$-dimensional faces of $Q_n$ is called $d$-polychromatic if every embedded $Q_d$ has every color on at least one face. Denote by $p^\ell(d)$ the maximum number of colors such that any $Q_n$ can be colored in this way. We provide a new lower bound on $p^\ell(d)$ for $\ell > 1$.