A hypergraph Turán theorem via lagrangians of intersecting families

Abstract

Let $\mc{K}_{3,3}^3$ be the 3-graph with 15 vertices $\{x_i,y_i:1\le i\le 3\}$ and $\{z_{ij}:1\le i,j\le 3\}$, and 11 edges $\{x_1,x_2,x_3\}$, $\{y_1,y_2,y_3\}$ and $\{\{x_i,y_j,z_{ij}\}:1\le i,j\le 3\}$. We show that for large $n$, the unique largest $\mc{K}_{3,3}^3$-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.