# The number of hypergraphs without linear cycles

## TL;DR

This paper establishes the asymptotic number of hypergraphs without certain linear cycles, confirming a conjecture and advancing understanding of hypergraph cycle structures.

## Contribution

It proves a balanced supersaturation result for linear cycles and applies hypergraph container methods to determine the count of cycle-free hypergraphs.

## Key findings

- Number of $C^r_k$-free hypergraphs is $2^{ heta(n^{r-1})}$.
- Settles a conjecture by Mubayi and Wang.
- Advances enumeration of hypergraphs with forbidden linear cycles.

## Abstract

The $r$-uniform linear $k$-cycle $C^r_k$ is the $r$-uniform hypergraph on $k(r-1)$ vertices whose edges are sets of $r$ consecutive vertices in a cyclic ordering of the vertex set chosen in such a way that every pair of consecutive edges share exactly one vertex. Here, we prove a balanced supersaturation result for linear cycles which we then use in conjunction with the method of hypergraph containers to show that for any fixed pair of integers $r, k \ge 3$, the number of $C^r_k$-free $r$-uniform hypergraphs on $n$ vertices is $2^{\Theta(n^{r-1})}$, thereby settling a conjecture due to Mubayi and Wang.