On strengthenings of the intersecting shadow theorem

Abstract

Let n>k>t≥j≥1 be integers. Let X be an n-element set, (Xk) the collection of its k-subsets. A family F⊂(Xk) is called t-intersecting if |F∩F′|≥t for all F,F′∈F. The j'th shadow ∂jF is the collection of all (k−j)-subsets that are contained in some member of F. Estimating |∂jF| as a function of |F| is a widely used tool in extremal set theory. A classical result of the second author (Theorem 1.3) provides such a bound for t-intersecting families. It is best possible for |F|=(2k−tk).Our main result is Theorem 1.4 which gives an asymptotically optimal bound on |∂jF|/|F| for |F| slightly larger, e.g., |F|>32(2k−tk). We provide further improvements for |F| very large as well.