Turan Problems on Non-uniform Hypergraphs

TL;DR

This paper extends Turán density concepts to non-uniform hypergraphs, exploring properties, degenerate cases, and fully determining densities for hypergraphs with edges of size 1 and 2, linking to extremal poset problems.

Contribution

It generalizes Turán density to non-uniform hypergraphs, analyzes properties like supersaturation and blow-up, and completely characterizes densities for 1-2 hypergraphs.

Findings

Generalized Turán density properties to non-uniform hypergraphs.

Identified which hypergraphs are degenerate in this context.

Determined Turán densities for all 1,2-hypergraphs.

Abstract

A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set of edge types. For a fixed hypergraph $H$, the Tur\'an density $\pi(H)$ is defined to be $\lim_{n\to\infty}\max_{G_n}h_n(G_n)$, where the maximum is taken over all $H$-free hypergraphs $G_n$ on $n$ vertices satisfying $R(G_n)\subseteq R(H)$, and $h_n(G_n)$, the so called Lubell function, is the expected number of edges in $G_n$ hit by a random full chain. This concept, which generalizes the Tur\'an density of $k$-uniform hypergraphs, is motivated by recent work on extremal poset problems. The details connecting these two areas will be revealed in the end of this paper. Several properties of Tur\'an density, such as supersaturation, blow-up, and suspension, are generalized from uniform hypergraphs to non-uniform hypergraphs. Other questions such as "Which hypergraphs are degenerate?" are more complicated and don't appear to generalize well. In addition, we completely determine the Tur\'an densities of ${1,2}$-hypergraphs.