Many neighborly spheres

TL;DR

This paper improves lower bounds on the number of distinct neighborly simplicial spheres with a given number of vertices, demonstrating exponential growth in the number of types as vertices increase.

Contribution

It provides a new construction that establishes significantly higher lower bounds for the number of neighborly spheres in dimensions five and above.

Findings

Exponential lower bounds for the number of neighborly spheres in higher dimensions.

Construction method for generating many distinct neighborly spheres.

Enhanced understanding of the combinatorial complexity of simplicial spheres.

Abstract

The result of Padrol asserts that for every $d\geq 4$, 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 $d\geq 5$, there are at least $2^{\Omega(n^{\lfloor (d-1)/2\rfloor})}$ such types.