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.