Neighborly polytope

Summary

A convex polytope is $k$-neighborly when every set of $k$ of its vertices spans a face. A $d$-polytope is called neighborly when it is $\floor{d/2}$-neighborly, the largest value possible for a polytope that is not a simplex. Neighborly polytopes exist in every dimension and with any number of vertices beyond $d$. Neighborly simplicial polytopes have the largest possible numbers of faces, which is why they appear in the upper bound theorem.