Families with restricted matching number and multiply covered shadows

Abstract

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|≤|L(n,k,s)| for s≥2k and n≥24k3s2, where L(n,k,s)={F⊂X:|F|=k,|F∩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.