Abstract Chvatal's conjecture on the intersecting family of the faces of the simplicial complex is a long-standing problem in combinatorics. Snevily gave an affirmative answer to this conjecture for near-cone complex. Woodroofe gave Erdős-Ko-Rado type theorem for near-cone complex by using algebraic shift method. Motivated by these results, we concern with the restricted intersecting family for the simplicial complex. First, we give an upper bound for the cardinality of the restricted intersecting family of the faces of the simplicial complex. Furthermore, we prove that if L = { l 1 , l 2 , … , l s } is a set of s positive integers, suppose that ▵ is a near-cone simplicial complex with an apex vertex v and F = { F 1 , … , F m } is a family of the faces of ▵ such that | F i ∩ F j | ∈ L for every 1 ≤ i ≠ j ≤ m , then m ≤ ∑ i = − 1 s − 1 f i ( link △ ( v ) ) , which generalizes Snevily's two theorems. We also propose a conjecture that this upper bound holds for all simplicial complexes. Finally, applying these theorems to certain simplicial complex, we can deduce the upper bounds for the cardinalities of the restricted intersecting families of the independent set of the graph, the set partition, the r-separated sets and the King Arthur and his Knight Table.

中文翻译：

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单纯复形上的受限相交族

摘要 Chvatal关于单纯复形面相交族的猜想是组合数学中的一个长期存在的问题。Snevily 对近锥络合物的这一猜想给出了肯定的回答。Woodroofe 使用代数移位方法给出了近锥复形的 Erdős-Ko-Rado 型定理。受这些结果的启发，我们关注单纯复形的受限相交族。首先，我们给出单纯复形面的受限相交族的基数上限。此外，我们证明如果 L = { l 1 , l 2 , … , ls } 是一组 s 个正整数，假设 ▵ 是一个近锥单纯复形，顶点为 v 且 F = { F 1 , … , F m } 是 ▵ 的面族，使得 | F i ∩ F j | ∈ L 对于每 1 ≤ i ≠ j ≤ m ，然后 m ≤ ∑ i = − 1 s − 1 fi ( link △ ( v ) ) ，它概括了 Sevily 的两个定理。我们还提出了一个猜想，即这个上限适用于所有单纯复形。最后，将这些定理应用于某个单纯复形，我们可以推导出图的独立集合、集合划分、r-分离集合和亚瑟王和他的骑士表的受限相交族的基数的上界。