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

Let be an n -uniform hypergraph on 2 n vertices. Suppose that | F 1 ∩ F 2 ∩ F 3 ∩ F 4 | ≥ 2 and | F 1 ∪ F 2 ∪ F 3 ∪ F 4 | ≤ n − 2 hold for all F 1 , F 2 , F 3 , F 4 ∈ . We prove that the size of is at most 2 n − 4 n − 2 for n sufficiently large.