In 1995, Jockusch constructed an infinite family of centrally symmetric $3$-dimensional simplicial spheres that are cs-$2$-neighborly. Here we generalize his construction and show that for all $d\geq 3$ and $n\geq d+1$, there exists a centrally symmetric $d$-dimensional simplicial sphere with $2n$ vertices that is cs-$\lceil d/2\rceil$-neighborly. This result combined with work of Adin and Stanley completely resolves the upper bound problem for centrally symmetric simplicial spheres.

中文翻译：

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高度邻域中心对称球体

1995 年，Jockusch 构造了一个无限的中心对称 $3$-维单纯球族，它们是 cs-$2$-邻域。在这里，我们推广他的构造，并证明对于所有 $d\geq 3$ 和 $n\geq d+1$，存在一个中心对称的 $d$ 维单纯球体，其顶点为 $2n$，即 cs-$\lceil d/2\rceil$-邻居。该结果与 Adin 和 Stanley 的工作相结合，完全解决了中心对称单纯球体的上限问题。