On the number of neighborly simplices in R^d

Abstract

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