A hypergraph Turán theorem via lagrangians of intersecting families

Let K 3,3 be the 3-graph with 15 vertices {xi, yi : 1 ≤ i ≤ 3} and {zij : 1 ≤ i, j ≤ 3}, and 11 edges {x1, x2, x3}, {y1, y2, y3} and {{xi, yj , zij} : 1 ≤ i, j ≤ 3}. We show that for large n, the unique largest K 3,3-free 3-graph on n vertices is a balanced blow-up of the complete 3-graph on 5 vertices. Our proof uses the stability method and a result on lagrangians of intersecting families that has independent interest.