| Abstrakt: |
A family Δ of subsets of { 1 , 2 , … , n } is a simplicial complex if all subsets of F are in Δ for any F ∈ Δ , and the element of Δ is called the face of Δ. Let V (Δ) = ⋃ F ∈ Δ F. A simplicial complex Δ is a near-cone with respect to an apex vertex v ∈ V (Δ) if for every face F ∈ Δ , the set (F \ { w }) ∪ { v } is also a face of Δ for every w ∈ F. Denote by f i (Δ) = | { A ∈ Δ : | A | = i + 1 } | and h i (Δ) = | { A ∈ Δ : | A | = i + 1 , n ∉ A } | for every i, and let link Δ (v) = { E : E ∪ { v } ∈ Δ , v ∉ E } for every v ∈ V (Δ). Assume that p is a prime and k ⩾ 2 is an integer. In this paper, some extremal problems on k-wise L-intersecting families for simplicial complexes are considered. (i) Let L = { l 1 , l 2 , … , l s } be a subset of s nonnegative integers. If F = { F 1 , F 2 , … , F m } is a family of faces of the simplicial complex Δ such that | F i 1 ∩ F i 2 ∩ ⋯ ∩ F i k | ∈ L for any collection of k distinct sets from F , then m ⩽ (k - 1) ∑ i = - 1 s - 1 f i (Δ). In addition, if the size of every member of F belongs to the set K : = { k 1 , k 2 , … , k r } with min K > s - r , then m ⩽ (k - 1) ∑ i = s - r s - 1 f i (Δ). (ii) Let L = { l 1 , l 2 , … , l s } and K = { k 1 , k 2 , … , k r } be two disjoint subsets of { 0 , 1 , … , p - 1 } such that min K > s - 2 r + 1. Assume that Δ is a simplicial complex with n ∈ V (Δ) and F = { F 1 , F 2 , … , F m } is a family of faces of Δ such that | F j | (mod p) ∈ K for every j and | F i 1 ∩ F i 2 ∩ ⋯ ∩ F i k | (mod p) ∈ L for any collection of k distinct sets from F. Then m ⩽ (k - 1) ∑ i = s - 2 r s - 1 h i (Δ). (iii) Let L = { l 1 , l 2 , … , l s } be a subset of { 0 , 1 , … , p - 1 }. Assume that Δ is a near-cone with apex vertex v and F = { F 1 , F 2 , … , F m } is a family of faces of Δ such that | F j | (mod p) ∉ L for every j and | F i 1 ∩ F i 2 ∩ ⋯ ∩ F i k | (mod p) ∈ L for any collection of k distinct sets from F. Then m ⩽ (k - 1) ∑ i = - 1 s - 1 f i (link Δ (v)). [ABSTRACT FROM AUTHOR] |
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