A hypergraph Turلn theorem via lagrangians of intersecting families

Let K 3 , 3 3 be the 3-graph with 15 vertices { x i , y i : 1 ⩽ i ⩽ 3 } and { z i j : 1 ⩽ i , j ⩽ 3 } , and 11 edges { x 1 , x 2 , x 3 } , { y 1 , y 2 , y 3 } and { { x i , y j , z i j } : 1 ⩽ i , j ⩽ 3 } . We show that for large n, the unique largest K 3 , 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.