On k-Wise L-Intersecting Families for Simplicial Complexes

A family Delta of subsets of {1,2,...,n} is a simplicial complex if all subsets of F are in Delta for any F is an element of Delta and the element of Delta is called the face of Delta. Let V(Delta) = boolean OR(F is an element of Delta) F. A simplicial complex Delta is a near-cone with respect to an apex vertex nu = (Delta) if for every face F is an element of Delta the set (F\{w}) boolean OR {v} is also a face of Delta for every omega is an element of F. Denote by f(i) (Delta) =|{A is an element of Delta : |A| = i + 1 } | and h(i )(Delta) = | { A is an element of Delta: |A| = i + 1, n /is an element of A} | for every i, and let link(Delta(v) )= {E : E boolean OR {v} is an element of Delta, v /is an element of E} for every v is an element of V(Delta) 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 Delta such that | F-i1 boolean AND F-i2 boolean AND <middle dot><middle dot><middle dot> boolean AND F-ik| is an element of L for any collection of k distinct sets from F then m <= (k-1) & sum;(s-1 )(i = -1)f(i)(Delta). 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)& sum;(s-1 )(i=s-r)f(i)(Delta). (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 - 2r + 1. Assume that Delta is a simplicial complex with n is an element of V(Delta)and F = {F-1,F-2,..., F-m} is a family of faces of Delta such that |F-j| (mod p) is an element of K for every j and |F-i1 boolean AND F-i2 boolean AND <middle dot><middle dot><middle dot> boolean AND F-ik| (mod p) is an element of L for any collection of k distinct sets from F. Then m <= (k-1)& sum;(s-1 )(i=s-2r)h(i)(Delta) (iii) Let L = {l(1),l(2),...,l(s)} be a subset of {0, 1, ..., p - 1} Assume that Delta is a near-cone with apex vertex v and F = {F-1,F-2 ,..., F-m} is a family of faces of Delta such that |F-j| (mod p) /is an element of L for every j and |F-i1 boolean AND F-i2 boolean AND<middle dot><middle dot><middle dot>boolean AND F-ik| (mod p) /is an element of L for any collection of k distinct sets from F .Then m <= (k-1) & sum;(s-1 )(i=-1)f(i )(link(Delta)(v)).