A hypergraph Turán theorem via lagrangians of intersecting families

Let K3,33 be the 3-graph with 15 vertices {xi,yi:1⩽i⩽3} and {zij:1⩽i,j⩽3}, and 11 edges {x1,x2,x3}{x1,x2,x3}, {y1,y2,y3}{y1,y2,y3} and {{xi,yj,zij}:1⩽i,j⩽3}. We show that for large n  , the unique largest K3,33-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.