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 | F | ⩽ ( n − t k − t ) if | k n − 1 2 | < ε holds for some ε > 0 and all n > n 0 ( ε ) . We apply this result to get EKR type inequalities for “intersecting and union families” and “intersecting Sperner families.”