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 $m\leqslant (k-1)\sum _{i=-1}^{s-1}f_i(\text {link}_\Delta (v)).$