The maximum size of 3-wise -intersecting families

Let t ≥ 26 and let ℱ be a k -uniform hypergraph on n vertices. Suppose that | F 1 ∩ F 2 ∩ F 3 | ≥ t holds for all F 1 , F 2 , F 3 ∈ ℱ . We prove that the size of ℱ is at most n − t k − t if p = k n satisfies p ≤ 2 4 t + 9 − 1 and n is sufficiently large. The above inequality for p is the best possible.