There are at most $2^{d+1}-2$ neighbourly simplices in dimension $d$

TL;DR

This paper proves a combinatorial theorem that establishes an upper bound of $2^{d+1}-2$ on the number of neighbourly simplices in d-dimensional space, using properties of disjoint sub-boxes in a discrete cube.

Contribution

It introduces a new combinatorial theorem linking disjoint sub-boxes in a discrete cube to the maximum number of neighbourly simplices in Euclidean space.

Findings

Maximum of $2^{d+1}-2$ neighbourly simplices in dimension d.

Establishes a connection between discrete cube sub-boxes and geometric simplices.

Provides a combinatorial proof for the upper bound.

Abstract

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