No more than $2^{d+1}-2$ nearly neighbourly simplices in $\mathbb R^d$

TL;DR

This paper establishes an upper bound on the number of nearly neighbourly simplices in d-dimensional space, using a combinatorial theorem related to families of disjoint sub-boxes of a discrete cube.

Contribution

It introduces a new combinatorial theorem that bounds the maximum number of nearly neighbourly simplices in Euclidean space.

Findings

Maximum of $2^{d+1}-2$ nearly neighbourly simplices in $ extbf{R}^d$

New combinatorial approach to bounding simplices

Implications for geometric and combinatorial theory

Abstract

We prove a combinatorial theorem on families of disjoint sub-boxes of a discrete cube, which implies that there are at most $2^{d+1}-2$ nearly neighbourly simplices in $\mathbb R^d$.