Random walks and multiply intersecting families

Let F ⊂ 2 [ n ] be a 3-wise 2-intersecting Sperner family. It is proved that | F | ⩽ n - 2 ( n - 2 ) / 2 if n even , n - 2 ( n - 1 ) / 2 + 2 if n odd holds for n ⩾ n 0 . The unique extremal configuration is determined as well.