# Improved Upper Bounds for Slicing the Hypercube

arXiv: https://arxiv.org/abs/2602.16807 · HTML: https://mlsanity.com/paper/2602.16807/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 39 references that are in the map.
- First posted on arXiv 6 Feb 2026.

## Builds on (most cited first, 15 of 15)

- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Mathematical exploration and discovery at scale](https://mlsanity.com/paper/2511.02864/) (2025)
- [Barbarians at the Gate: How AI is Upending Systems Research](https://mlsanity.com/paper/2510.06189/) (2025)
- [CodeEvolve: an open source evolutionary coding agent for algorithmic discovery and optimization](https://mlsanity.com/paper/2510.14150/) (2025)
- [Early science acceleration experiments with GPT-5](https://mlsanity.com/paper/2511.16072/) (2025)
- [PatternBoost: Constructions in Mathematics with a Little Help from AI](https://mlsanity.com/paper/2411.00566/) (2024)
- [Can Language Models Discover Scaling Laws?](https://mlsanity.com/paper/2507.21184/) (2025)
- [Advancing Geometry with AI: Multi-agent Generation of Polytopes](https://mlsanity.com/paper/2502.05199/) (2025)
- [Finding Increasingly Large Extremal Graphs with AlphaZero and Tabu Search](https://mlsanity.com/paper/2311.03583/) (2023)
- [NuMVC: An Efficient Local Search Algorithm for Minimum Vertex Cover](https://mlsanity.com/paper/1402.0584/) (2014)
- [Using Code Generation to Solve Open Instances of Combinatorial Design Problems](https://mlsanity.com/paper/2501.17725/) (2025)
- [Using Reasoning Models to Generate Search Heuristics that Solve Open Instances of Combinatorial Design Problems](https://mlsanity.com/paper/2505.23881/) (2025)
- [Global Optimization for Combinatorial Geometry Problems Revisited in the Era of LLMs](https://mlsanity.com/paper/2601.05943/) (2026)
- [Flow-based Extremal Mathematical Structure Discovery](https://mlsanity.com/paper/2601.18005/) (2026)
- [Slicing the hypercube is not easy](https://mlsanity.com/paper/2102.05536/) (2021)

## Built on by (most cited first, 1 of 1)

- [Automated Discovery of Improved Constant Weight Binary Codes](https://mlsanity.com/paper/2603.00174/) (2026)

## Abstract (arXiv)

A collection of hyperplanes H slices all edges of the n-dimensional hypercube Qn with vertex set -1,1n if, for every edge e in the hypercube, there exists a hyperplane in H intersecting e in its interior. Let S(n) be the minimum number of hyperplanes needed to slice Qn. We prove that S(n) ≤ lceil frac4n5 rceil, except when n is an odd multiple of 5, in which case S(n) ≤ frac4n5 +1. This improves upon the previously known upper bound of S(n) ≤ lceilfrac5n6 rceil due to Paterson reported in 1971. We also obtain new lower bounds on the maximum number of edges in Qn that can be sliced using k<n hyperplanes. We prove the improved upper bound on S(n) by constructing 8 hyperplanes slicing Q10 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.