A recent conjecture that appeared in three papers by Bigdeli–Faridi, Dochtermann, and Nikseresht, is that every simplicial complex whose clique complex has shellable Alexander dual, is ridge-chordal. This strengthens the long-standing Simon's conjecture that the k-skeleton of the simplex is extendably shellable, for any k. We show that the stronger conjecture has a negative answer, by exhibiting an infinite family of counterexamples.

中文翻译：

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非脊弦和弦复合体，其派系复合体具有可轰击的亚历山大二重体

Bigdeli–Faridi，Dochtermann和Nikseresht在三篇论文中最近提出的一个猜想是，其派系具有可剥壳的亚历山大二重性的每个单纯形复杂体都是脊脊弦的。这加强了长期以来的西蒙猜想，即对于任何k，单纯形的k骨架都可扩展地可炮轰。通过展示无限的反例族，我们证明了更强的猜想有一个否定的答案。