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Improved Upper Bounds for Slicing the Hypercube

Duncan Soiffer, Nathaniel Itty, Christopher D. Rosin, Blake Bruell, Mason DiCicco, Gábor N. Sárközy, Ryan Offstein, Daniel Reichman

Abstract

A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its interior. Let $S(n)$ be the minimum number of hyperplanes needed to slice $Q_n$. We prove that $S(n) \leq \lceil \frac{4n}{5} \rceil$, except when $n$ is an odd multiple of $5$, in which case $S(n) \leq \frac{4n}{5} +1$. This improves upon the previously known upper bound of $S(n) \leq \lceil\frac{5n}{6} \rceil$ due to Paterson reported in 1971. We also obtain new lower bounds on the maximum number of edges in $Q_n$ that can be sliced using $k<n$ hyperplanes. We prove the improved upper bound on $S(n)$ by constructing $8$ hyperplanes slicing $Q_{10}$ aided by the recently introduced CPro1: an automatic tool that uses reasoning LLMs coupled with automated hyperparameter tuning to create search algorithms for the discovery of mathematical constructions.

Improved Upper Bounds for Slicing the Hypercube

Abstract

A collection of hyperplanes slices all edges of the -dimensional hypercube with vertex set if, for every edge in the hypercube, there exists a hyperplane in intersecting in its interior. Let be the minimum number of hyperplanes needed to slice . We prove that , except when is an odd multiple of , in which case . This improves upon the previously known upper bound of due to Paterson reported in 1971. We also obtain new lower bounds on the maximum number of edges in that can be sliced using hyperplanes. We prove the improved upper bound on by constructing hyperplanes slicing aided by the recently introduced CPro1: an automatic tool that uses reasoning LLMs coupled with automated hyperparameter tuning to create search algorithms for the discovery of mathematical constructions.
Paper Structure (17 sections, 6 theorems, 16 equations, 1 figure, 6 tables, 2 algorithms)

This paper contains 17 sections, 6 theorems, 16 equations, 1 figure, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Related work
  4. Algorithms for Slicing the Hypercube
  5. Algorithm Search
  6. Local search
  7. Reduced Hypercube
  8. AI-Driven Discovery and Researcher Intervention
  9. Discussion and Conclusions
  10. Paterson construction for slicing $Q_6$
  11. Additional constructions for slicing $Q_{10}$
  12. CPro1's initial improvement to S(15,12)
  13. The Reduced Hypercube
  14. Computing the Sliced Edges in the Reduced Hypercube
  15. Additional Tables
  16. ...and 2 more sections

Key Result

Theorem 1

For every $n$ such that $n$ is not an odd multiple of $5$, $S(n) \leq \lceil\frac{4n}{5} \rceil$. Otherwise, $S(n) \leq \frac{4n}{5}+1$.

Figures (1)

  • Figure 1: A $6$-dimensional hypercube under the $[3,2,1]$ composition reduces to 46 edges, and is transformed into a $3\times2\times1$ lattice in three dimensions.

Theorems & Definitions (17)

  • Theorem 1
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 2
  • proof
  • Corollary 1
  • Definition 5
  • ...and 7 more