### References & Citations

# Mathematics > Combinatorics

# Title: Counting Hypergraphs with Large Girth

(Submitted on 4 Oct 2020 (v1), last revised 16 Oct 2021 (this version, v2))

Abstract: Morris and Saxton used the method of containers to bound the number of $n$-vertex graphs with $m$ edges containing no $\ell$-cycles, and hence graphs of girth more than $\ell$. We consider a generalization to $r$-uniform hypergraphs. The {\em girth} of a hypergraph $H$ is the minimum $\ell$ such that for some $F \subseteq H$, there exists a bijection $\phi : E(C_\ell) \to E(F)$ with $e\subseteq \phi(e)$ for all $e\in E(C_\ell)$. Letting $N_m^r(n,\ell)$ denote the number of $n$-vertex $r$-uniform hypergraphs with $m$ edges and girth larger than $\ell$ and defining $\lambda = \lceil (r - 2)/(\ell - 2)\rfloor$, we show \[ N_m^r(n,\ell) \leq N_m^2(n,\ell)^{r - 1 + \lambda}\] which is tight when $\ell - 2 $ divides $r - 2$ up to a $1 + o(1)$ term in the exponent. This result is used to address the extremal problem for subgraphs of girth more than $\ell$ in random $r$-uniform hypergraphs.

## Submission history

From: Sam Spiro [view email]**[v1]**Sun, 4 Oct 2020 03:58:28 GMT (47kb,D)

**[v2]**Sat, 16 Oct 2021 22:51:28 GMT (49kb,D)

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