We prove that the family of facets of a pure simplicial complex $\mathcal{C}$ of dimension up to three satisfies the Erdős-Ko-Rado property whenever $\mathcal{C}$ is flag and has no boundary ridges. We conjecture the same to be true in arbitrary dimension and give evidence for this conjecture. Our motivation is that complexes with these two properties include flag pseudo-manifolds and cluster complexes.

中文翻译：

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EKR属性，用于标记无边界的纯单纯形复形

我们证明纯单纯复形的面族 $\mathcal{C}$ 尺寸的三分之一满足Erdős-Ko-Rado属性 $\mathcal{C}$是标志，没有边界脊。我们猜想在任意维度上都是一样的，并为此作证。我们的动机是具有这两个属性的复合体包括标志伪流形和簇复合体。