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

M. R. Emamy-K.1, R. Arce-Nazario2, L. J. Uribe3
1 Dept. of Mathematics, University of Puerto Rico, Rio Piedras,PR, USA
2Dept. of Computer Science, University of Puerto Rico, Rio Piedras,PR, USA
3Dept. of Edu. Technology, Boise State University, Idaho, USA

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 \( d \leq 4 \) 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.

Keywords: Hypercube, Hyperplanes, Cube Cuts CONGRESSUS NUMERANTIUM 232 (2019)