Many neighborly spheres

Abstract

The result of Padrol asserts that for every $d\ge 4$, there exist $2^{\Omega (n\log n)}$ distinct combinatorial types of $\lfloor d/2\rfloor$-neighborly simplicial $(d-1)$-spheres with $n$ vertices. We present a construction showing that for every $d\ge 5$, there are at least $2^{\Omega (n^{\lfloor (d-1)/2\rfloor })}$ such types.

Rephrased Definition 3.3 and added Definition 3.8; added more details, including proof details, several examples, and an outline of the proof of Theorem 3.1. To appear in Mathematische Annalen