Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.02329

The Boolean lattice $\mathcal{B}_6$ is not a sphere order

David A. Meyer
View original source ↗Read source document ↗Export metadata ↓

Abstract

Using the equivalence of sphere order representations of posets with Minkowski space embeddings, we show that the Boolean lattice $\mathcal{B}_6$ is not a sphere order. This is the first explicit finite example of a poset that is not a sphere order.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

David A. Meyer. 2026-10-01. The Boolean lattice $\mathcal{B}_6$ is not a sphere order. https://arxiv.org/abs/2610.02329

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Symmetries of the q-deformed real projective line

We generalize in two steps the quantized action of the modular group on q-deformed real numbers introduced by Morier-Genoud and Ovsienko. First, we let the projective general linear group PGL(2,Z) act on q-real numbers via a q-deformed action. The deformed matrices we get have combinatorial interpretations, and we show that their traces are palindromic polynomials. Then we consider an extension of the group PGL(2,Z) by the 2-elements cyclic group, and define a deformed action of this extension on q-real numbers. We deduce from these actions some underlying relations between q-real numbers, and between left and right versions of q-deformed rational numbers. In particular we investigate the case of some algebraic numbers of degree 4 and 6. We also prove that the deformation of real numbers is an injective process.

math.CO↗
arXiv2026

Combinatorial Games and the Golden Ratio on Digraphs

We introduce a new combinatorial game called Triangle Game. In this game, a directed $3$-cycle graph is given, and stones are placed on each vertex. The player chooses a directed edge and takes at least one stone from the initial vertex. At the same time, the player is allowed to return some stones to the terminal vertex of the edge, as long as the total number of stones decreases. We describe the set of \Pps~under both normal play and misère play. The golden ratio $ϕ=\dfrac{1+\sqrt{5}}{2}$ plays an essential role in our description. We also show that Triangle Game is tame.

math.CO↗
arXiv2026

Coloopless zonotopes and counterexamples to the Shifted Lonely Runner Conjecture

Henze and Malikiosis (2017) have shown that the Lonely Runner Conjecture (LRC) can be restated as a convex-geometric question on the so-called LR zonotopes, lattice zonotopes with one more generator than their dimension. This relation naturally suggests a more general statement, the shifted LRC, the zonotopal version of which concerns a classical parameter, the covering radius. In this paper we do two things: 1) We show explicit counterexamples to both the shifted Lonely Runner Conjecture (starting at $n=5$) and to the Lonely Vector Property of Malikiosis-Schymura-Santos (2025). 2) We push the analogies between the two versions of LRC and their zonotopal counterparts, in particular highlighting that the proofs of the finite-checking Theorems A and B in Malikiosis-Schymura-Santos (2025) are more transparent, and the statements more general, if regarded in terms of two quite general classes of lattice zonotopes: the coloopless zonotopes that we introduce here and the cosimple ones, already defined by them.

math.CO↗