On the Tur\'an density of $\{1, 3\}$-Hypergraphs

TL;DR

This paper investigates Turán problems for hypergraphs with edges of size 1 or 3, characterizing degeneracy via colorability, and computes Turán densities for specific small hypergraphs.

Contribution

It establishes a characterization of degeneracy for ,3-hypergraphs using a specific colorability criterion and computes Turán densities for certain small hypergraphs.

Findings

A ,3-hypergraph is degenerate iff it is $H^{\u0011,3}_5$-colorable.

Degenerate $R$-graphs exist for all finite sets $R$ of positive integers except .

Turán densities for some small ,3-hypergraphs are explicitly computed.

Abstract

In this paper, we consider the Tur\'an problems on $\{1,3\}$-hypergraphs. We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set $V=[5]$ and edge set $E=\{\{2\}, \{3\}, \{1, 2, 4\}, \{1, 3, 5\}, \{1, 4, 5\}\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\{1, 2\}$, there always exist non-trivial degenerate $R$-graphs. We also compute the Tur\'an densities of some small $\{1,3\}$-hypergraphs.