By considering the number of maximal chains going through each element of an arbitrary poset, we prove an extension of Erdős’s generalisation of Sperner’s Theorem, together with a partial converse. By considering the number of maximal chains between pairs of comparable elements, we also prove a generalisation of the LYM inequality.

中文翻译：

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通过他们的最大链调查posets

通过考虑通过任意偏序集的每个元素的最大链的数量，我们证明了 Erdős 对 Sperner 定理的推广以及部分逆推论的扩展。通过考虑可比元素对之间的最大链数，我们还证明了 LYM 不等式的推广。