Families with restricted matching number and multiply covered shadows

Let X be an n-element set and let F be a family consisting of k-subsets of X. The matching number of F is defined as the maximum number of pairwise disjoint members in F. A (k -1)-set E is called a shadow of F if E is contained in some member of F. The minimum shadow-degree of F is defined as the minimum integer d such that every shadow E of F is contained in at least d members of F. In the present paper, we show that if F has matching number at most s and minimum shadow-degree at least s + 1, then |F| <= |.C(n, k, s)| for s >= 2k and n >= 24k3s2, where .C(n, k, s) = {F subset of X : |F| = k, |F boolean AND Y| >= 2} with Y being a (2s + 1)-subset of X. We prove the analogous statement for s = 2, 3 and k arbitrary as well as k = 3 and n >= 20s + 36. (c) 2023 Elsevier B.V. All rights reserved.