On k-Wise L-Intersecting Families for Simplicial Complexes

Abstract

A family \(\Delta \) of subsets of \(\{1,2,\ldots ,n\}\) is a simplicial complex if all subsets of F are in \(\Delta \) for any \(F\in \Delta ,\) and the element of \(\Delta \) is called the face of \(\Delta .\) Let \(V(\Delta )=\bigcup _{F\in \Delta } F.\) A simplicial complex \(\Delta \) is a near-cone with respect to an apex vertex \(v\in V(\Delta )\) if for every face \(F\in \Delta ,\) the set \((F\backslash \{w\})\cup \{v\}\) is also a face of \(\Delta \) for every \(w\in F.\) Denote by \(f_{i}(\Delta )=|\{A\in \Delta :|A|=i+1\}|\) and \(h_{i}(\Delta )=|\{A\in \Delta :|A|=i+1,n\not \in A\}|\) for every i, and let \(\text {link}_{\Delta }(v)=\{E:E\cup \{v\}\in \Delta , v\not \in E\}\) for every \(v\in V(\Delta ).\) Assume that p is a prime and \(k\geqslant 2\) is an integer. In this paper, some extremal problems on k-wise L-intersecting families for simplicial complexes are considered. (i) Let \(L=\{l_1,l_2,\ldots ,l_s\}\) be a subset of s nonnegative integers. If \(\mathscr {F}=\{F_1, F_2,\ldots , F_m\}\) is a family of faces of the simplicial complex \(\Delta \) such that \(|F_{i_1}\cap F_{i_2}\cap \cdots \cap F_{i_k}|\in L\) for any collection of k distinct sets from \(\mathscr {F},\) then \(m\leqslant (k-1)\sum _{i=-1}^{s-1}f_i(\Delta ).\) In addition, if the size of every member of \(\mathscr {F}\) belongs to the set \(K:=\{k_1,k_2,\ldots ,k_r\}\) with \(\min K>s-r,\) then \(m\leqslant (k-1)\sum _{i=s-r}^{s-1}f_i(\Delta ).\) (ii) Let \(L=\{l_1,l_2,\ldots ,l_s\}\) and \(K=\{k_1,k_2,\ldots ,k_r\}\) be two disjoint subsets of \(\{0,1,\ldots ,p-1\}\) such that \(\min K>s-2r+1.\) Assume that \(\Delta \) is a simplicial complex with \(n\in V(\Delta )\) and \(\mathscr {F}=\{F_1, F_2,\ldots , F_m\}\) is a family of faces of \(\Delta \) such that \(|F_j|\pmod {p}\in K\) for every j and \(|F_{i_1}\cap F_{i_2}\cap \cdots \cap F_{i_k}|\pmod {p}\in L\) for any collection of k distinct sets from \(\mathscr {F}.\) Then \(m\leqslant (k-1)\sum _{i=s-2r}^{s-1}h_i(\Delta ).\) (iii) Let \(L=\{l_1,l_2,\ldots ,l_s\}\) be a subset of \(\{0,1,\ldots ,p-1\}.\) Assume that \(\Delta \) is a near-cone with apex vertex v and \(\mathscr {F}=\{F_1, F_2,\ldots , F_m\}\) is a family of faces of \(\Delta \) such that \(|F_j|\pmod {p}\not \in L\) for every j and \(|F_{i_1}\cap F_{i_2}\cap \cdots \cap F_{i_k}|\pmod {p}\in L\) for any collection of k distinct sets from \(\mathscr {F}.\) Then \( m\leqslant (k-1)\sum _{i=-1}^{s-1}f_i(\text {link}_\Delta (v)).\)