On Shadows Of Intersecting Families

The shadow minimization problem for t-intersecting systems of finite sets is considered. Let $${\user1{A}}$$ be a family of k-subsets of ℕ. The l-shadow of $${\user1{A}}$$ is the set of all (k-l)-subsets $$\partial _{{\ell }} {\user1{A}}$$ contained in the members of $${\user1{A}}$$. Let $${\user1{A}}$$ be a t-intersecting family (any two members have at least t elements in common) with $${\left| {\user1{A}} \right|} = m$$. Given k,t,m the problem is to minimize $${\left| {\partial _{{\ell }} {\user1{A}}} \right|}$$ (over all choices of $${\user1{A}}$$). In this paper we solve this problem when m is big enough.