EKR type inequalities for 4-wise intersecting families

Let 1⩽t⩽7 be an integer and let F be a k-uniform hypergraph on n vertices. Suppose that |A∩B∩C∩D|⩾t holds for all A,B,C,D∈F. Then we have if holds for some ε>0 and all n>n0(ε). We apply this result to get EKR type inequalities for “intersecting and union families” and “intersecting Sperner families.”