Kusner's taxicab equilateral-set conjecture, first open case: is $e(\ell_1^5)=10$?
Statement
Work in $\mathbb{R}^n$ with the $\ell_1$ (taxicab / Manhattan) metric $\|x-y\|_1=\sum_{i=1}^{n}|x_i-y_i|$. An equilateral set is a set of points at pairwise equal $\ell_1$-distance; write $e(\ell_1^n)$ for the maximum cardinality of such a set. Kusner conjectured $e(\ell_1^n)=2n$, attained by $\{\pm e_1,\dots,\pm e_n\}$. This is the $\ell_1$ (that is, $p=1$) Kusner conjecture, with conjectured value $2n$ - distinct from the $\ell_p$ ($p>1$) conjecture, whose conjectured value is $n+1$. The first open case is $n=5$: decide whether $e(\ell_1^5)=10$ (i.e. $=2\cdot5$), equivalently whether $11$ points of $\mathbb{R}^5$ can be pairwise $\ell_1$-equidistant.
Acceptance. FULLY RESOLVES: EITHER explicit coordinates of $11$ pairwise $\ell_1$-equidistant points in $\mathbb{R}^5$ (a rational, exactly-checkable yes-certificate disproving the conjecture at $n=5$), OR a proof that no $11$ such points exist - e.g. an exhaustive, certificate-bearing case analysis over the finite family of coordinate sign/order patterns establishing $e(\ell_1^5)=10$. ADVANCES: a verified reduction of the sign-pattern search space for $n=5$; the analogous decision for $n=6$; or an improved general upper bound below $O(n\log n)$. Floating-point-only near-equidistant configurations do NOT qualify.
Background
Kusner's conjecture (posed ~1983, reported by Guy). The $\ell_1$ value $2n$ is confirmed proven only for $n\le4$: $n=1,2$ are elementary, $n=3$ is due to Bandelt-Chepoi-Laurent (1998), and $n=4$ to Koolen-Laurent-Schrijver ('Equilateral dimension of the rectilinear space', 2000). The general upper bound is $e(\ell_1^n)=O(n\log n)$ (Alon-Pudlak, 'Equilateral sets in $\ell_p^n$'). GOLD SIGNAL: H.-J. Ge, Z. Xu, Y. Zhou, 'Kusner's conjecture: Exact values and linear bounds', arXiv:2606.03987 (June 2026), states in its introduction that $e(\ell_1^n)=2n$ is 'known only ... for $n\le4$', so $n=5$ is the first open case. That paper proves the $\ell_p$ conjecture for $2\le p\le4$ and disproves it for $1<p<2$, but it does NOT resolve the $p=1$ taxicab case, which is a separate older question (a reader skimming its abstract could wrongly think the taxicab case moved). $\ell_1$-equilateral sets carry strong sign / coordinate-pattern constraints, so '11 pairwise $\ell_1$-equidistant points in $\mathbb{R}^5$' recasts as a finite feasibility problem over the $\{+,-,0\}$ sign patterns of coordinate differences (mixed-integer / SAT). Vetted open as of 2026-07-06 (high confidence; June-2026 gold signal).
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Ge, Xu & Zhou, Kusner's conjecture: Exact values and linear bounds (June 2026) - e(l_1^n)=2n known only for n<=4 | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.