Linear d-Polychromatic $$Q_{d-1}$$ Q d - 1 -Colorings of the Hypercube

Abstract

Let $$n \ge d \ge \ell \ge 1$$ be integers, and denote the n-dimensional hypercube by $$Q_n$$ . A coloring of the $$\ell $$ -dimensional subcubes $$Q_\ell $$ in $$Q_n$$ is called a $$Q_\ell $$ -coloring. Such a coloring is d-polychromatic if every $$Q_d$$ in the $$Q_n$$ contains a $$Q_\ell $$ of every color. In this paper we consider a specific class of $$Q_\ell $$ -colorings that are called linear. Given $$\ell $$ and d, let $$p_{lin}^\ell (d)$$ be the largest number of colors such that there is a d-polychromatic linear $$Q_\ell $$ -coloring of $$Q_n$$ for all $$n \ge d$$ . We prove that for all $$d \ge 3$$ , $$p_{lin}^{d-1}(d) = 2$$ . In addition, using a computer search, we determine $$p_{lin}^\ell (d)$$ for some specific values of $$\ell $$ and d, in some cases improving on previously known lower bounds.