Turan Problems and Shadows III: expansions of graphs

TL;DR

This paper investigates the maximum size of 3-uniform hypergraphs avoiding certain expanded graphs, revealing new bounds and characterizing when this maximum is quadratic in the number of vertices.

Contribution

It establishes asymptotic bounds for the extremal function of expanded bipartite graphs and characterizes graphs with quadratic extremal functions.

Findings

For complete bipartite graphs, the extremal number is (n^{3 - 3/s}) when t > (s - 1)! and s  3.

Graphs with Tur number o(n^{\u03c6}) have quadratic extremal functions.

The extremal function for the 3D cube graph is (n^2).

Abstract

The expansion $G^+$ of a graph $G$ is the $3$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a new vertex disjoint from $V(G)$ such that distinct edges are enlarged by distinct vertices. Let $ex_3(n,F)$ denote the maximum number of edges in a $3$-uniform hypergraph with $n$ vertices not containing any copy of a $3$-uniform hypergraph $F$. The study of $ex_3(n,G^+)$ includes some well-researched problems, including the case that $F$ consists of $k$ disjoint edges, $G$ is a triangle, $G$ is a path or cycle, and $G$ is a tree. In this paper we initiate a broader study of the behavior of $ex_3(n,G^+)$. Specifically, we show \[ ex_3(n,K_{s,t}^+) = \Theta(n^{3 - 3/s})\] whenever $t > (s - 1)!$ and $s \geq 3$. One of the main open problems is to determine for which graphs $G$ the quantity $ex_3(n,G^+)$ is quadratic in $n$. We show that this occurs when $G$ is any bipartite graph with Tur\'{a}n number $o(n^{\varphi})$ where $\varphi = \frac{1 + \sqrt{5}}{2}$, and in particular, this shows $ex_3(n,Q^+) = \Theta(n^2)$ where $Q$ is the three-dimensional cube graph.