A hypergraph Tur\'an theorem via lagrangians of intersecting families

TL;DR

This paper determines the maximum size of a 3-uniform hypergraph avoiding a specific subgraph, using stability methods and lagrangian techniques related to intersecting families.

Contribution

It establishes the unique extremal structure for large hypergraphs avoiding a particular configuration, introducing new lagrangian bounds for intersecting families.

Findings

Identifies the extremal 3-graph avoiding rac{3,3}^3.

Proves the extremal graph is a balanced blow-up of a 5-vertex complete 3-graph.

Develops new lagrangian bounds for 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.