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

Abstract

AbstractTwo d-dimensional simplices in $$\mathbb {R}^d$$ R d are neighborly if its intersection is a $$(d-1)$$ ( d - 1 ) -dimensional set. A family of d-dimensional simplices in $$\mathbb {R}^d$$ 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 $$\mathbb {R}^d$$ R d . Based on the structure of some codes $$V\subset \{0,1,*\}^n$$ V ⊂ { 0 , 1 , ∗ } n it is shown that $$\lim _{d\rightarrow \infty }(2^{d+1}-S_d)=\infty $$ lim d → ∞ ( 2 d + 1 - S d ) = ∞ . Moreover, a result on the structure of codes $$V\subset \{0,1,*\}^n$$ V ⊂ { 0 , 1 , ∗ } n is given.