The maximum size of 3-wise tt-intersecting families

Let t≥26t≥26 and let ℱℱ be a kk-uniform hypergraph on nn vertices. Suppose that |F1∩F2∩F3|≥t|F1∩F2∩F3|≥t holds for all F1,F2,F3∈ℱF1,F2,F3∈ℱ. We prove that the size of ℱℱ is at most n−tk−t if p=kn satisfies p≤24t+9−1 and nn is sufficiently large. The above inequality for pp is the best possible.