Improved Upper Bounds for Slicing the Hypercube
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arXiv
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| Autores principales: | , , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910026596089856 |
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| author | Soiffer, Duncan Itty, Nathaniel Rosin, Christopher D. Bruell, Blake DiCicco, Mason Sárközy, Gábor N. Offstein, Ryan Reichman, Daniel |
| author_facet | Soiffer, Duncan Itty, Nathaniel Rosin, Christopher D. Bruell, Blake DiCicco, Mason Sárközy, Gábor N. Offstein, Ryan Reichman, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16807 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Improved Upper Bounds for Slicing the Hypercube Soiffer, Duncan Itty, Nathaniel Rosin, Christopher D. Bruell, Blake DiCicco, Mason Sárközy, Gábor N. Offstein, Ryan Reichman, Daniel Artificial Intelligence Discrete Mathematics Combinatorics 05-04 G.2.1; I.2.2 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. |
| title | Improved Upper Bounds for Slicing the Hypercube |
| topic | Artificial Intelligence Discrete Mathematics Combinatorics 05-04 G.2.1; I.2.2 |
| url | https://arxiv.org/abs/2602.16807 |