The 5-cube Cut Number Problem: A Short Proof for a Basic Lemma

Abstract

The hypercube cut number $S(d)$ is the minimum number of hyperplanes in the $d$-dimensional Euclidean space ${\mathbb{R}}^d$ that slice all the edges of the $d$-cube. The problem was originally posed by P. O’Neil in 1971. B. Grünbaum, V. Klee, M. Saks, and Z. Füredi have raised the problem in various contexts.

The identity $S(d)=d$ has been well-known for $d\le 4$ since 1986. However, it was only until the year 2000 that Sohler and Ziegler obtained a computational proof for $S(5)=5$. Nevertheless, finding a short proof for the problem, independent of computer computations, remains a challenging task.

We present a short proof for the result presented by Emamy-Uribe-Tomassini in Hypercube 2002 based on Tomassini’s Thesis. The proof here is substantially shorter than the original proof of 60 pages.