On the number of linear uniform hypergraphs with linear girth constraint

TL;DR

This paper investigates the number of linear r-uniform hypergraphs with large girth, providing new lower bounds using probabilistic methods and analyzing the random greedy process.

Contribution

It introduces probabilistic deletion techniques and analyzes the hypergraph process to establish improved lower bounds on the count of hypergraphs with specified girth constraints.

Findings

Established lower bounds for the number of hypergraphs with large girth

Analyzed the random greedy process for linear hypergraphs

Extended results to various r and girth parameters

Abstract

For an integer $r\geqslant 3$, a hypergraph on vertex set $[n]$ is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if every two distinct edges share at most one vertex. Given a family $\mathcal{H}$ of linear $r$-uniform hypergraphs,let $Forb_r^L(n,\mathcal{H})$ be the set of linear $r$-uniform hypergraphs on vertex set $[n]$, which does not contain any member from $\mathcal{H}$ as a subgraph. An $r$-uniform linear cycle of length $\ell$, denoted by $C_\ell^r$, is a linear $r$-uniform hypergraph on $(r-1)\ell$ vertices whose edges can be ordered as $\boldsymbol{e}_1,\ldots,\boldsymbol{e}_\ell$ such that $|\boldsymbol{e}_i\cap \boldsymbol{e}_j|=1$ if $j=i\pm 1$ (indices taken modulo $\ell$) and $|\boldsymbol{e}_i\cap \boldsymbol{e}_j|=0$ otherwise. The linear girth of a linear $r$-uniform hypergraph is the smallest integer $\ell$ such that it contains a $C_\ell^r$. Let $Forb_L(n,r,\ell)=Forb_r^L(n,\mathcal{H})$ when $\mathcal{H}=\{C_i^r:\, 3\leqslant i\leqslant \ell\}$, that is, $Forb_L(n,r,\ell)$ is the set of all linear $r$-uniform hypergraphs on $[n]$ with linear girth greater than $\ell$. For integers $r\geqslant 3$ and $\ell\geqslant 4$, Balogh and Li [On the number of linear hypergraphs of large girth, J. Graph Theory, 93(1) (2020), 113-141] showed that $|Forb_L(n,r,\ell)|= 2^{O(n^{1+1/\lfloor \ell/2\rfloor})}$ based on the graph container method. It is natural to obtain $|Forb_L(n,r,\ell)|\geqslant 2^{c\cdot n^{1+1/\ell}}$ for some constant $c$ by probabilistic deletion method. Combined with the known results that $|Forb_L(n,r,3)|= 2^{o (n^{2})}$ and $|Forb_L(n,3,4)|= 2^{\Theta (n^{3/2})}$, by analyzing the random greedy high linear girth linear $r$-uniform hypergraph process, we show $|Forb_L(n,r,\ell)|\geqslant 2^{n^{1+1/(\ell-1)-O(\log\log n/\log n)}}$ for every pair of fixed integers $r,\ell\geqslant 4$, or $r= 3$ and $\ell\geqslant 5$.