On strengthenings of the intersecting shadow theorem

Abstract Let n > k > t ≥ j ≥ 1 be integers. Let X be an n-element set, ( X k ) the collection of its k-subsets. A family F ⊂ ( X k ) is called t-intersecting if | F ∩ F ′ | ≥ t for all F , F ′ ∈ F . The j'th shadow ∂ j F is the collection of all ( k − j ) -subsets that are contained in some member of F . Estimating | ∂ j F | 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 | = ( 2 k − t k ) . Our main result is Theorem 1.4 which gives an asymptotically optimal bound on | ∂ j F | / | F | for | F | slightly larger, e.g., | F | > 3 2 ( 2 k − t k ) . We provide further improvements for | F | very large as well.