We consider the question of the largest possible combinatorial diameter among $(d-1)$-dimensional simplicial complexes on $n$ vertices, denoted $H_s(n, d)$. Using a probabilistic construction we give a new lower bound on $H_s(n, d)$ that is within an $O(d^2)$ factor of the upper bound. This improves on the previously best-known lower bound which was within a factor of $e^{\Theta(d)}$ of the upper bound. We also make a similar improvement in the case of pseudomanifolds.

中文翻译：

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大直径配合物的随机构建

我们考虑$n$顶点上$(d-1)$维单纯复形中最大可能组合直径的问题，记为$H_s(n, d)$。使用概率构造，我们给出了 $H_s(n, d)$ 的新下限，该下限在上限的 $O(d^2)$ 因子内。这改进了之前最著名的下限，该下限在上限的 $e^{\Theta(d)}$ 因子内。我们也在伪流形的情况下进行了类似的改进。