The maximum size of 4-wise 2-intersecting and 4-wise 2-union families

Let be an nn-uniform hypergraph on 2n2n vertices. Suppose that |F1∩F2∩F3∩F4|≥2|F1∩F2∩F3∩F4|≥2 and |F1∪F2∪F3∪F4|≤n−2|F1∪F2∪F3∪F4|≤n−2 hold for all F1,F2,F3,F4∈F1,F2,F3,F4∈. We prove that the size of is at most 2n−4n−2 for nn sufficiently large.