Sub-25-dimensional counterexamples to Borsuk's conjecture in the Leech lattice?

TL;DR

The work probes Borsuk's conjecture in sub-25 dimensions by embedding the problem in the Leech lattice $\Lambda\subset\mathbb{R}^{24}$ and studying the finite sets $M$ of norm-$\sqrt{32}$ vectors and the associated choice-sets $H$. It translates partitioning into a graph-coloring problem on the inner-product graph with threshold $-16$ and uses the TABUCOL heuristic to obtain minimal colorings for laminated sublattices $\Lambda_n$, yielding $M_n$ and $H_n$ partitions. The main findings show that $M_n$ remains partitionable into at most $n+1$ parts for $n\le 21$, but for $n=22,23,24$ the observed minima are $25$, $29$, and $34$ parts, respectively, with a concrete $H_{24}$ instance requiring at least $29$ parts, constituting a potential sub-25D counterexample. The results are computational, delivering explicit data and a reproducible candidate set to facilitate verification and further exploration of Borsuk's conjecture in higher dimensions.

Abstract

In 1933, Karol Borsuk asked whether each bounded set in the $n$-dimensional Euclidean space can be divided into $n$+1 parts of smaller diameter. Because it would not make sense otherwise, one usually assumes that he just forgot to require that the whole set contains at least two points. The hypothesis that the answer to that question is positive became famous under the name \emph{Borsuk's conjecture}. Counterexamples are known for any $n\ge 64$, since 2013. Let $Λ$ be the (original, unscaled) Leech lattice, a now very well-known infinite discrete vector set in the 24-dimensional Euclidean space. The smallest norm of nonzero vectors in $Λ$ is $\sqrt{32}$. Let $M$ be the set of the 196560 vectors in $Λ$ having this norm. For each $x \in M$, $-x$ is in $M$. Let $H$ be the set of all subsets of $M$ that for each $x$ in $M$ contain either $x$ or $-x$. Each element of $H$ has the same diameter $d = \sqrt{96}$. For dimensions $n<24$ one can analogously construct respective $M_n$ and $H_n$ from laminated $n$-dimensional sublattices $Λ_n$ of $Λ$. For uniformity, let $Λ_{24}=Λ$, $M_{24} = M$ and $H_{24} = H$. If $M_n$ is divisible into at most $n+1$ parts of diameter below $d$ then this applies to all elements of $H_n$, too. I have checked that this is the case for all $n \le 21$. For $n$ from 22 to 24, the minimum number of parts of diameter below $d$ that I was able to divide $M_n$ into are 25, 29 and 34, resp. The source package of this article contains a data file encoding an element of $H_{24}=H$ that I can not divide into less than 29 parts of smaller diameter.