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 vertical bar A boolean AND B boolean AND C boolean AND D vertical bar >= t holds for all A, B, C, D is an element of F. Then we have vertical bar F <= ((n-t)(k-t)) if vertical bar k/n - 1/2 vertical bar < epsilon holds for some epsilon > 0 and all n > n(9)(epsilon). We apply this result to get EKR type inequalities for "intersecting and union families" and "intersecting Sperner families." (c) 2006 Elsevier Inc. All rights reserved.