# Asymptotics for Tur\'an numbers of cycles in 3-uniform linear hypergraphs

## TL;DR

This paper determines the asymptotic maximum edge counts in 3-uniform linear hypergraphs avoiding certain cycles, revealing precise growth rates and connections to graph girth and extremal numbers.

## Contribution

It establishes the asymptotic linear Turán numbers for cycles of length 4, 5, and certain odd lengths in 3-uniform linear hypergraphs, linking hypergraph extremal problems to graph girth.

## Key findings

- Linear Turán number of C5 is asymptotically (1/3√3) n^{3/2}.
- Linear Turán numbers of C4 and {C3, C4} are asymptotically equal.
- Linear Turán number of odd cycles of length 2k+1 is Θ(n^{1+1/k}) for specific k.

## Abstract

Let $\mathcal{F}$ be a family of $3$-uniform linear hypergraphs. The linear Tur\'an number of $\mathcal F$ is the maximum possible number of edges in a $3$-uniform linear hypergraph on $n$ vertices which contains no member of $\mathcal{F}$ as a subhypergraph. In this paper we show that the linear Tur\'an number of the five cycle $C_5$ (in the Berge sense) is $\frac{1}{3 \sqrt3}n^{3/2}$ asymptotically. We also show that the linear Tur\'an number of the four cycle $C_4$ and $\{C_3, C_4\}$ are equal asmptotically, which is a strengthening of a theorem of Lazebnik and Verstra\"ete. We establish a connection between the linear Tur\'an number of the linear cycle of length $2k+1$ and the extremal number of edges in a graph of girth more than $2k-2$. Combining our result and a theorem of Collier-Cartaino, Graber and Jiang, we obtain that the linear Tur\'an number of the linear cycle of length $2k+1$ is $\Theta(n^{1+\frac{1}{k}})$ for $k = 2, 3, 4, 6$.