On the Number of Neighborly Simplices in \(\mathbb {R}^d\)

Two d-dimensional simplices in \(\mathbb {R}^d\) ${\mathbb{R}}^d$ are neighborly if its intersection is a \((d-1)\) $(d-1)$ -dimensional set. A family of d-dimensional simplices in \(\mathbb {R}^d\) ${\mathbb{R}}^d$ is called neighborly if every two simplices of the family are neighborly. Let \(S_d\) $S_d$ be the maximal cardinality of a neighborly family of d-dimensional simplices in \(\mathbb {R}^d\) ${\mathbb{R}}^d$ . Based on the structure of some codes \(V\subset \{0,1,*\}^n\) $V\subset {\{0,1,\ast \}}^n$ it is shown that \(\lim _{d\rightarrow \infty }(2^{d+1}-S_d)=\infty \) ${lim}_{d\to \infty }(2^{d+1}-S_d)=\infty$ . Moreover, a result on the structure of codes \(V\subset \{0,1,*\}^n\) $V\subset {\{0,1,\ast \}}^n$ is given.