A hypergraph Turán theorem via Lagrangians of intersecting families

Abstract

Let $K_{3,3}^3$, be the 3-graph with 15 vertices {x _i , y _i :1 ≤ i ≤ 3} and {z _ij :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 _ij }: 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.