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\begin{document}
\title{On regular hypergraphs of high girth}

\author{David Ellis\thanks{Research supported in part by a Feinberg Visiting Fellowship from the Weizmann Institute of Science.}\\
\small School of Mathematical Sciences\\[-0.8ex]
\small Queen Mary, University of London\\[-0.8ex] 
\small UK\\
\small\tt d.ellis@qmul.ac.uk\\
\and
Nathan Linial\\
\small School of Computer Science and Engineering\\[-0.8ex]
\small Hebrew University of Jerusalem\\[-0.8ex]
\small Israel\\
\small\tt nati@cs.huji.ac.il
}

% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{Nov 1, 2013}{}\\
\small Mathematics Subject Classifications: 05C65}
\maketitle

\begin{abstract}
We give lower bounds on the maximum possible girth of an $r$-uniform, $d$-regular hypergraph with at most $n$ vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between $3/2+o(1)$ and $2 +o(1)$). We also define a random $r$-uniform `Cayley' hypergraph on the symmetric group $S_n$ which has girth $\Omega (\sqrt{\log |S_n|})$ with high probability, in contrast to random regular $r$-uniform hypergraphs, which have constant girth with positive probability.
\end{abstract}

\section{Introduction}
The {\em girth} of a finite graph $G$ is the shortest length of a cycle in $G$. (If $G$ is acyclic, we define its girth to be $\infty$.) The {\em girth problem} asks for the minimum possible number of vertices $n(g,d)$ in a $d$-regular graph of girth at least $g$, for each pair of integers $d,g \geq 3$. Equivalently, for each pair of integers $n,d \geq 3$ with $nd$ even, it asks for a determination of the largest possible girth $g_{d}(n)$ of a $d$-regular graph on at most $n$ vertices.

The girth problem has received much attention for more than half a century, starting with Erd\H{o}s and Sachs \cite{erdossachs}. A fairly easy probabilistic argument shows that for any integers $d,g \geq 3$, there exist $d$-regular graphs with girth at least $g$. An extremal argument due to Erd\H{o}s and Sachs \cite{erdossachs} then shows that there exists such a graph with at most
$$2\frac{(d-1)^{g-1}-1}{d-2}$$
vertices. This implies that
\begin{equation}\label{eq:erdos-sachs}g_{d}(n) \geq (1- o(1)) \log_{d-1}n.\end{equation}
(Here, and below, $o(1)$ stands for a function of $n$ that tends to zero as $n \to \infty$.)

On the other hand, if $G$ is a $d$-regular graph of girth at least $g$, then counting the number of vertices of $G$ of distance less than $g/2$ from a fixed vertex of $G$ (when $g$ is odd), or from a fixed edge of $G$ (when $G$ is even), immediately shows that
$$|G| \geq n_0(g,d) := \left\{\begin{array}{lll} 1+d\sum_{i=0}^{k-1}(d-1)^i & = 1+d\frac{(d-1)^{k}-1}{d-2} & \textrm{if }g =2k+1;\\ 2\sum_{i=0}^{k-1}(d-1)^i & = 2\frac{(d-1)^{k}-1}{d-2} & \textrm{if }g=2k.\end{array}\right.$$
This is known as the {\em Moore bound}. Graphs for which the Moore bound holds with equality are known as {\em Moore graphs} (for odd $g$), or {\em generalized polygons} (for even $g$). It is known that Moore graphs only exist when $g=3$ or $5$, and generalized polygons only exist when $g = 4,6,8$ or $12$. It was proved in \cite{bannai,biggs,kovacs} that if $d \geq 3$, then
$$n(g,d) \geq n_0(g,d)+2\quad \textrm{for all } g \notin \{3,4,5,6,8,12\};$$
even for large values of $g$ and $d$, no improvement on this is known.

A related problem is to give an explicit construction of a $d$-regular graph of girth $g$, with as few vertices as possible. The celebrated Ramanujan graphs constructed by Lubotzsky, Phillips and Sarnak \cite{lps}, Margulis \cite{margulis} and Morgenstern \cite{morgenstern} constituted a breakthrough on both problems, implying that
\begin{equation}\label{eq:lps} g_{d}(n) \geq (4/3 - o(1))\log_{d-1}n\end{equation}
via an explicit (algebraic) construction, whenever $d =q+1$ for some odd prime power $q$.

One can obtain from this a lower bound on $g_{d}(n)$ for arbitrary $d \geq 3$, by choosing the minimum $d' \geq d$ such that $d' - 1$ is an odd prime power, taking a $d'$-regular Ramanujan graph with girth achieving (\ref{eq:lps}), and removing $d'-d$ perfect matchings in succession. This yields 
\begin{equation}\label{eq:lps2} g_{d}(n) \geq (4/3 - o(1))\frac{\log(d-1)}{\log(d'-1)} \log_{d-1}n.\end{equation}

In \cite{luw1} and \cite{luw2}, Lazebik, Ustimenko and Woldar give different explicit constructions (also algebraic), which imply that 
$$g_{d}(n) \geq (4/3 - o(1))\log_{d}n$$
whenever $d$ is an odd prime power, implying (\ref{eq:lps2}) whenever $d-1$ is not an odd prime power. (In fact, their constructions provide the best known upper bound on $n(g,d)$ for many pairs of values $(g,d)$.) Combining (\ref{eq:lps2}) with the Moore bound gives
\begin{equation}\label{eq:upper} (4/3 - o(1))\frac{\log(d-1)}{\log(d'-1)} \log_{d-1}n \leq g_{d}(n) \leq (2+o(1)) \log_{d-1}n.\end{equation}
Improving the constants in (\ref{eq:upper}) seems to be a very hard problem.

In this paper, we investigate an analogue of the girth problem for $r$-uniform hypergraphs, where $r \geq 3$. There are several natural notions of a cycle in a hypergraph. We refer the reader to Section~\ref{sec:open} for a brief discussion of some other interesting notions of girth in hypergraphs, and to \cite{duke} for a detailed treatise. Here, we consider the least restrictive notion, originally due to Berge (see for example \cite{berge} and \cite{berge2}). \begin{comment} Namely, if $l \geq 2$, a cycle of length $l$ is a sequence of $l$ edges $(e_1,\ldots,e_l)$  such that there exist $l$ distinct vertices $v_1,\ldots,v_l$ with $v_i \in e_i \cap e_{i+1}$ for all $i$ (where we define $e_{l+1} := e_1$). \end{comment}

A {\em hypergraph} $H$ is a pair of finite sets $(V(H),E(H))$, where $E(H)$ is a family of subsets of $V(H)$. The elements of $V(H)$ are called the {\em vertices} of $H$, and the elements of $E(H)$ are called the {\em edges} of $H$. A hypergraph is said to be {\em $r$-uniform} if all its edges have size $r$. It is said to be {\em $d$-regular} if each of its vertices is contained in exactly $d$ edges. It is said to be {\em linear} if any two of its edges share at most one vertex.

Let $u$ and $v$ be distinct vertices in a hypergraph $H$. A {\em u-v path} of {\em length} $l$ in $H$ is a sequence of distinct edges $(e_1,\ldots,e_l)$ of $H$, such that $u \in e_1$, $v \in e_l$, $e_i \cap e_{i+1} \neq \emptyset$ for all $i \in \{1,2,\ldots,l-1\}$, and $e_i \cap e_j = \emptyset$ whenever $j > i+1$ (Note that some authors call this a  {\em geodesic path}, and use the term {\em path} when non-consecutive edges are allowed to intersect.) The {\em distance} from $u$ to $v$ in $H$, denoted $\textrm{dist}(u,v)$, is the shortest length of a {\em u-v} path in $H$. (We define $\dist(v,v) = 0$.) The {\em ball of radius $R$ and centre $u$} in $H$ is the set of vertices of $H$ with distance at most $R$ from $u$. The {\em diameter} of a hypergraph $H$ is defined by
$$\diam(H) = \max_{u,v \in V(H)} \dist(u,v).$$

A hypergraph is said to be a {\em cycle} if it has at least two edges, and there is a cyclic ordering of its edges, $(e_1,\ldots,e_l)$ say, such that there exist distinct vertices $v_1,\ldots,v_l$ with $v_i \in e_i \cap e_{i+1}$ for all $i$ (where we define $e_{l+1} := e_1$). This notion of a hypergraph cycle is originally due to Berge, and is sometimes called a {\em Berge-cycle}. The {\em length} of a cycle is the number of edges in it. The {\em girth} of a hypergraph is the length of the shortest cycle it contains.

Observe that two distinct edges $e,f$ with $|e \cap f| \geq 2$ form a cycle of length 2 under this definition, so when considering hypergraphs of high girth, we may restrict our attention to linear hypergraphs.

We use the Landau notation for functions: if $F,G:\mathbb{N} \to \mathbb{R}^{+}$, we write $F = o(G)$ if $F(n)/G(n) \to 0$ as $n \to \infty$. We write $F = O(G)$ if there exists $C>0$ such that $F(n) \leq CG(n)$ for all $n$. We write $F = \Omega(G)$ if there exists $c>0$ such that $F(n) \geq cG(n)$ for all $n$. Finally, we write $F = \Theta(G)$ if $F = O(G)$ and $F = \Omega(G)$.

Extremal questions concerning Berge-cycles in hypergraphs have been studied by several authors. For example, in \cite{bg}, Bollob\'as and Gy\H{o}ri prove that an $n$-vertex, 3-uniform hypergraph with no 5-cycle has at most $\sqrt{2} n^{3/2} + \tfrac{9}{2}n$ edges, and they give a construction showing that this is best possible up to a constant factor. In \cite{lv}, Lazebnik and Verstra\"ete prove that a 3-uniform, $n$-vertex hypergraph of girth at least 5 has at most
$$\tfrac{1}{6}n \sqrt{n-\tfrac{3}{4}} + \tfrac{1}{12}n$$
edges, and give a beautiful construction (based on the so-called `polarity graph' of the projective plane $\textrm{PG}(2,q)$) showing that this is sharp whenever $n = q^2$ for an odd prime power $q \geq 27$. Interestingly, neither of these two constructions are regular.

In \cite{gl} and \cite{thesis}, Gy\"ori and Lemons consider the problem of excluding a cycle of length exactly $k$, for general $k \in \mathbb{N}$. In \cite{gl}, they prove that an $n$-vertex, 3-uniform hypergraph with no $(2k+1)$-cycle has at most $4k^2 n^{1+1/k}+O(n)$ edges. In \cite{thesis}, they prove that an $n$-vertex, $r$-uniform hypergraph with no $(2k+1)$-cycle has at most $C_{k,r}(n^{1+1/k})$ edges, and furthermore that an $n$-vertex, $r$-uniform hypergraph with no $(2k)$-cycle has at most $C_{k,r}'(n^{1+1/k})$ edges, where $C_{k,r}, C_{k,r}'$ depend upon $k$ and $r$ alone.

In this paper, we will investigate the maximum possible girth of an $r$-uniform, $d$-regular hypergraph on $n$ vertices, for $r$ and $d$ fixed and $n$ large. If $r \geq 3$ and $d \geq 2$, we let $g_{r,d}(n)$ denote the maximum possible girth of an $r$-uniform, $d$-regular hypergraph on at most $n$ vertices. Similarly, if $d \geq 2$ and $r,g \geq 3$, we let $n_r(g,d)$ denote the minimum possible number of vertices in an $r$-uniform, $d$-regular hypergraph with girth at least $g$. Since a non-linear hypergraph has girth 2, we may replace `hypergraph' with `linear hypergraph' in these two definitions.

In section 2, we will state upper and lower bounds on the function $g_{r,d}(n)$, which differ by an absolute constant factor. The upper bound is a simple analogue of the Moore bound for graphs, and follows immediately from known results. The lower bound is a hypergraph extension of a similar argument for graphs, due to Erd\H{o}s and Sachs \cite{erdossachs} --- not a particularly difficult extension, but still, in our opinion, worth recording.

In section 3, we consider the girth of certain kinds of random $r$-uniform hypergraph. We define a random $r$-uniform `Cayley' hypergraph on $S_n$ which has girth $\Omega (\sqrt{\log |S_n|})$ with high probability, in contrast to random regular $r$-uniform hypergraphs, which have constant girth with positive probability. We conjecture that, in fact, our `Cayley' hypergraph has girth $\Omega(\log |S_n|)$ with high probability. We believe it may find other applications.

\section{Upper and lower bounds}
In this section, we state upper and lower bounds on the function $g_{r,d}(n)$, which differ by an absolute constant factor.

We first state a very simple analogue of the Moore bound for linear hypergraphs. For completeness, we give the proof, although the result follows immediately from known results, e.g. from Theorem 1 of Hoory \cite{hoorybipartite}.
\begin{lemma}
Let $r,d$ and $g$ be integers with $d \geq 2$ and $r,g \geq 3$. Let $H$ be an $r$-uniform, $d$-regular, $n$-vertex hypergraph with girth $g$. If $g=2k+1$ is odd, then
\begin{equation}\label{eq:odd}n \geq 1+d(r-1)\sum_{i=0}^{k-1} ((d-1)(r-1))^i = 1+d(r-1)\frac{(d-1)^{k}(r-1)^{k}-1}{(d-1)(r-1)-1},\end{equation}
and if $g=2k$ is even, then
\begin{equation}\label{eq:even}n \geq r\sum_{i=0}^{k-1} ((d-1)(r-1))^i = r\frac{(d-1)^{k}(r-1)^{k}-1}{(d-1)(r-1)-1}.\end{equation}
\end{lemma}
\begin{proof}
The right-hand side of (\ref{eq:odd}) is the number of vertices in any ball of radius $k$. The right-hand side of (\ref{eq:even}) is the number of vertices of distance at most $k-1$ from any fixed edge $e \in H$.
\end{proof}
The following corollary is immediate.
\begin{corollary}
\label{corr:upperbound}
Let $r,d$ and $g$ be integers with $d \geq 2$ and $r,g \geq 3$. Let $H$ be an $r$-uniform, $d$-regular hypergraph with $n$ vertices and girth $g$. Then
$$g \leq \frac{2\log n}{\log(r-1)+\log(d-1)}+2.$$
Hence,
$$g_{r,d}(n) \leq  \frac{2\log n}{\log(r-1)+\log(d-1)}+2.$$
\end{corollary}

Our aim is now to obtain a hypergraph analogue of the non-constructive lower bound (\ref{eq:erdos-sachs}). We first prove the following existence lemma.

\begin{lemma}
\label{lemma:lift}
For all integers $d \geq 2$ and $r,g \geq 3$, there exists a finite, $r$-uniform, $d$-regular hypergraph with girth at least $g$.
\end{lemma}
\begin{proof}
We prove this by induction on $g$, for fixed $r,d$. When $g=3$, all we need is a linear, $r$-uniform, $d$-regular hypergraph. Let $H$ be the hypergraph on vertex-set $\mathbb{Z}_r^d$, whose edges are all the axis-parallel lines, i.e. 
$$E(H) = \{\{\mathbf{x},\mathbf{x}+\mathbf{e}_i,\mathbf{x}+2\mathbf{e}_i,\ldots,\mathbf{x}+(r-1)\mathbf{e}_i\}:\ \mathbf{x} \in \mathbb{Z}_r^d,\ i \in [d]\}.$$
(Here, $\mathbf{e}_i$ denotes the $i$th standard basis vector in $\mathbb{Z}_r^d$, i.e. the vector with $1$ in the $i$th coordinate and zero elsewhere. As usual, $\mathbb{Z}_r$ denotes the ring of integers modulo $r$.) Clearly, $H$ is linear and $d$-regular. 

For $g \geq 4$ we do the induction step. We start from a finite, linear, $r$-uniform, $d$-regular hypergraph $H$ of girth at least $g-1$. Of all such hypergraphs we consider one with the least possible number of $(g-1)$-cycles. Let $M$ be the number of $(g-1)$-cycles in $H$. We shall prove that $M=0$. If $M>0$, we consider a random 2-lift $H'$ of $H$, defined as follows. Its vertex set is $V(H') = V(H) \times \{0,1\}$, and its edges are defined as follows. For each edge $e \in E(H)$, choose an arbitrary ordering $(v_1,\ldots,v_r)$ of the vertices in $e$, flip $r-1$ independent fair coins $c^{(1)}_e,\ldots,c_e^{(r-1)} \in \{0,1\}$, and include in $H'$ the two edges
$$\{(v_1,j),(v_2,j \oplus c^{(1)}_e),\ldots,(v_r,j\oplus c^{(r-1)}_e)\}\mbox{~~for~~}j=0,1.$$
(Here, $\oplus$ denotes modulo 2 addition.) Do this independently for each edge. Note that $H'$ is linear and $d$-regular, since $H$ is.

Let $\pi:V(H') \to V(H)$ be the cover map, defined by $\pi((v,j))=v$ for all $v \in V(H)$ and $j \in \{0,1\}$. Since any cycle in $H'$ is projected to a cycle in $H$ of the same length, $H'$ has girth at least $g-1$, and each $(g-1)$-cycle in $H'$ projects to a $(g-1)$-cycle in $H$. Let $C$ be a $(g-1)$-cycle in $H$. We claim that $\pi^{-1}(C)$ either consists of two vertex-disjoint $(g-1)$-cycles in $H'$, or a single $2(g-1)$-cycle in $H'$, and that the probability of each is $1/2$. To see this, let $(e_1,\ldots,e_{g-1})$ be any cyclic ordering of $C$; then $|e_i \cap e_{i+1}|=1$ for all $i$ (since $H$ is linear). Let $e_i \cap e_{i+1} = \{w_i\}$ for all $i \in [g-1]$. For each $i$, consider the two edges in $\pi^{-1}(e_i)$. Either one of the two edges contains $(w_{i-1},0)$ and $(w_{i},0)$ and the other contains $(w_{i-1},1)$ and $(w_{i},1)$, {\em or} one edge contains $(w_{i-1},0)$ and $(w_i,1)$ and the other edge contains $(w_{i-1},1)$ and $(w_i,0)$. Call these two events $S(e_i)$ and $D(e_i)$, for `same' and `different'. Observe that $S(e_i)$ and $D(e_i)$ each occur with probability 1/2, independently for each edge $e_i$ in the cycle. Notice that $\pi^{-1}(C)$ consists of two disjoint $(g-1)$-cycles if and only if $D(e_i)$ occurs an even number of times, and the probability of this is $1/2$, proving the claim.

It follows that the expected number of $(g-1)$-cycles in $H'$ is $M$. Note that the trivial lift $H_0$ of $H$, which has $c^{(k)}_e = 0$ for all $k$ and $e$, consists of two vertex-disjoint copies of $H$, and therefore has $2M$ $(g-1)$-cycles. It follows that there is at least one 2-lift of $H$ with fewer than $M$ $(g-1)$-cycles, contradicting the minimality of $M$. Therefore, $M=0$, so in fact, $H$ has girth at least $g$. This completes the proof of the induction step, proving the theorem.
\end{proof}

\begin{remark} Lemma \ref{lemma:lift} can also be proved by considering a random $r$-uniform, $d$-regular hypergraph on $n$ vertices, for $n$ large. In \cite{cooper}, Cooper, Frieze, Molloy and Reed analyse these using a generalisation of Bollob\'as' configuration model for $d$-regular graphs. It follows from Lemma 2 in \cite{cooper} that if $H$ is chosen uniformly at random from the set of all $r$-uniform, $d$-regular, $n$-vertex, linear hypergraphs (where $r |n$), then
\begin{equation}\label{eq:randomreg} \textrm{Prob}\{\textrm{girth}(H) \geq g\} = (1+o(1)) \frac{\exp(-\sum_{l=1}^{g-1} \lambda_l)}{1-\exp(-(\lambda_1+\lambda_2))},\end{equation}
where
$$\lambda_i = \frac{(r-1)^i (d-1)^i}{2i} \quad (i \in \mathbb{N}),$$
so this event occurs with positive probability for sufficiently large $n$, giving an alternative proof of Lemma \ref{lemma:lift}. (We note that the argument of \cite{cooper} can easily be adapted to prove the same statement in the case where $r\, |\, dn$.) \end{remark}

By itself, the proof of Lemma \ref{lemma:lift} implies only that
$$n_r(g,d) \leq \underbrace{
  {{{2^{2\vphantom{h}}}^{2\vphantom{h}}}^{\iddots\vphantom{h}}}^{2^{r^{Cd}}\vphantom{h}}
}_{\text{$g-3\ 2$'s}},$$
where $C$ is an absolute constant --- i.e., tower-type dependence upon $g$. We now proceed to obtain an upper bound which is exponential in $g$.

Consider a $d$-regular graph with girth at least $g$, with the smallest possible number of vertices subject to these conditions. Erd\H{o}s and Sachs \cite{erdossachs} proved that the diameter of such a graph is at most $g$. But a $d$-regular graph with diameter $D$ has at most
$$1+d\sum_{i=0}^{D-1}(d-1)^i$$
vertices (since this is an upper bound on the number of vertices in a ball of radius $D$). This yielded the upper bound (\ref{eq:erdos-sachs}) on the number of vertices in a $d$-regular graph of girth at least $g$ and minimal order.

We need an analogue of the Erd\H{o}s-Sachs argument for hypergraphs.

\begin{lemma}
Let $r,d$ and $g$ be integers with $d \geq 2$ and $r,g \geq 3$. Let $H$ be an $r$-uniform, $d$-regular hypergraph with girth at least $g$, with the smallest possible number of vertices subject to these conditions. Then $H$ cannot contain $r$ vertices every two of which are at distance greater than $g$ from one another.
\end{lemma}
\begin{proof}
Let $H$ be an $r$-uniform, $d$-regular hypergraph with girth at least $g$. Suppose that $H$ contains $r$ distinct vertices $v_1,v_2,\ldots,v_r$ such that $\dist(v_i,v_j) > g$ for all $i \neq j$. We will show that it is then possible to construct an $r$-uniform, $d$-regular hypergraph with girth at least $g$, that has fewer vertices than $H$; this will prove the lemma.

Note that $H$ is linear, since $g \geq 3$. For each $i \in [r]$, let $e^{(1)}_i,e^{(2)}_i,\ldots,e^{(d)}_i$ be the edges of $H$ which contain $v_i$. Let
$$W_i = \bigcup_{k=1}^{d}(e^{(k)}_i \setminus \{v_i\})$$
for each $i \in [r]$. Notice that $|W_i| = d(r-1)$ for each $i$, since the edges $e^{(k)}_i\ (k \in [d])$ are disjoint apart from the vertex $v_i$. Moreover, $W_i \cap W_j = \emptyset$ for all $i \neq j$, since $d(v_i,v_j) > 2$.

Define a new hypergraph $H'$ by taking $H$, deleting $v_1,v_2,\ldots,v_r$ and all the edges containing them, and adding $d(r-1)$ pairwise disjoint edges, each of which contains exactly one vertex from $W_i$ for each $i \in [r]$. (Note that none of these `new' edges were in the original hypergraph $H$, otherwise some $v_{i}$ and $v_{j}$  would have been at distance at most 3 in $H$, a contradiction.) Clearly, $H'$ is $d$-regular. We claim that it is linear. Indeed, if one of the `new' edges shared two vertices with some edge $f \in H$ (say it shares $a \in W_i$ and $b \in W_j$, where $i \neq j$), then there would be a path of length 3 in $H$ from $v_i$ to $v_j$, a contradiction. 

We now claim that $H'$ has girth at least $g$. Suppose for a contradiction that $H'$ has girth at most $g-1$. Let $C$ be a cycle in $H'$ of length $l \leq g-1$. Since $H'$ is linear, we have $l \geq 3$. Let $(f_1,\ldots,f_l)$ be a cyclic ordering of $C$. We split into two cases.

\textit{Case 1.} Suppose that $C$ contains exactly one of the `new' edges (say $f_i$ is a `new' edge). Deleting $f_i$ from $C$ produces a path $P$ of length at most $g-2$ in $H$. We have $|f_{i-1} \cap f_i| = |f_i \cap f_{i+1}|=1$ (since $H'$ is linear); let $f_{i-1}\cap f_i = \{a\}$, and let $f_{i} \cap f_{i+1}=\{b\}$. Note that $a \neq b$. Suppose that $a \in W_p$ and $b \in W_q$. Since $a \neq b$ and $a,b \in f_i$, we must have $p \neq q$, as each `new' edge contains exactly one vertex from each $W_k$. Let $e$ be the edge of $H$ containing both $v_p$ and $a$, and let $e'$ be the edge of $H$ containing both $v_q$ and $b$; adding $e$ and $e'$ to the appropriate ends of the path $P$ produces a path in $H$ of length at most $g$ from $v_p$ to $v_q$, contradicting the assumption that $\textrm{dist}(v_p,v_q) > g$.

\textit{Case 2.} Suppose instead that $C$ contains more than one of the `new' edges. Choose a minimal sub-path $P$ of $C$ which connects two `new' edges. Suppose $P$ connects the new edges $f_i$ and $f_j$, so that $P = (f_i,f_{i+1},\ldots,f_{j-1},f_j)$. Note that $|i-j| \leq (g-1)/2$, so $P$ has length at most $(g+1)/2 \leq g-1$. Let $f_i \cap f_{i+1} = \{a\}$, and suppose $a \in W_p$; let $f_{j-1} \cap f_j = \{b\}$, and suppose $b \in W_q$. Let $e$ be the unique edge of $H$ which contains both $v_p$ and $a$, and let $e'$ be the unique edge of $H$ which contains both $v_q$ and $b$. If $p \neq q$, then we can produce a path in $H$ from $v_p$ to $v_q$ by taking $P$, and replacing $f_i$ with $e$ and $f_j$ with $e'$; this path has length at most $g-1$, contradicting our assumption that $d(v_p,v_q) > g$. If $p=q$, then we can produce a cycle in $H$ by taking $P$, removing $f_i$ and $f_j$, and adding the edges $e$ and $e'$ (which share the vertex $v_p$); this cycle has length at most $g-1$, contradicting our assumption that $H$ has girth at least $g$.

We may conclude that $H'$ has girth at least $g$, as claimed. Clearly, $H'$ has fewer vertices than $H$; this completes the proof.
\end{proof}

This lemma quickly implies an upper bound on the minimal number of vertices in an $r$-uniform, $d$-regular hypergraph of girth at least $g$.

\begin{theorem}
Let $r,d$ and $g$ be integers with $d \geq 2$ and $r,g \geq 3$. There exists an $r$-uniform, $d$-regular hypergraph with girth at least $g$, and at most
$$(r-1)\left(1+d(r-1)\frac{(d-1)^g (r-1)^g -1}{(d-1)(r-1)-1}\right) < 4((d-1)(r-1))^{g+1}$$
vertices. Hence,
$$n_r(g,d) < 4((d-1)(r-1))^{g+1}.$$
\end{theorem}
\begin{proof}
Let $H$ be an $r$-uniform, $d$-regular hypergraph with girth at least $g$, with the smallest possible number of vertices subject to these conditions. Let $\{v_1,v_2,\ldots,v_k\}$ be a set of vertices of $H$ whose pairwise distances are all greater than $g$, with $k$ maximal subject to this condition. By the previous lemma, we have $k < r$. Any vertex of $H$ must have distance at most $g$ from one of the $v_i$'s. For each $i$, the number of vertices of $H$ of distance at most $g$ from $v_i$ is at most
$$1+d(r-1)\sum_{i=0}^{g-1} ((d-1)(r-1))^i = 1+d(r-1) \frac{(d-1)^g (r-1)^g -1}{(d-1)(r-1)-1},$$
and therefore the number of vertices of $H$ is at most
$$k\left(1+d(r-1)\frac{(d-1)^g (r-1)^g -1}{(d-1)(r-1)-1}\right) \leq (r-1)\left(1+d(r-1)\frac{(d-1)^g (r-1)^g -1}{(d-1)(r-1)-1}\right).$$
Crudely, we have
$$(r-1)\left(1+d(r-1)\frac{(d-1)^g (r-1)^g -1}{(d-1)(r-1)-1}\right) < 4((d-1)(r-1))^{g+1}$$
for all integers $r,d$ and $g$ with $d \geq 2$ and $r,g \geq 3$, proving the theorem.
\end{proof}
The following corollary is immediate.
\begin{corollary}
\label{corr:lowerbound}
Let $r,d$ and $n$ be positive integers with $d \geq 2$ and $r \geq 3$. There exists an $r$-uniform, $d$-regular hypergraph on at most $n$ vertices, with girth greater than
$$ \frac{\log n - \log 4}{\log(d-1)+\log(r-1)} - 1.$$
Hence,
$$g_{r,d}(n) > \frac{\log n - \log 4}{\log(d-1)+\log(r-1)} - 1.$$
\end{corollary}

Observe that the lower bound in Corollary \ref{corr:lowerbound} differs from the upper bound in Corollary \ref{corr:upperbound} by a factor of (approximately) 2.

For $r,d \geq 3$, we have not been able to improve upon the lower bound in Corollary \ref{corr:lowerbound} for large $n$. As mentioned in the Introduction, in the case of graphs, the bipartite Ramanujan graphs of Lubotzsky, Phillips and Sarnak \cite{lps}, Margulis \cite{margulis} and Morgenstern \cite{morgenstern} provide $d$-regular, $n$-vertex graphs of girth at least
$$(1-o(1))\frac{4}{3} \frac{\log n}{\log (d-1)},$$
for infinitely many $n$, whenever $d-1$ is a prime power. Recall that a finite, connected, $d$-regular graph is said to be {\em Ramanujan} if every eigenvalue $\lambda$ of its adjacency matrix is either `trivial' (i.e. $\lambda = \pm d$), or has $|\lambda| \leq 2\sqrt{d-1}$.

\begin{theorem}[Lubotzsky-Phillips-Sarnak, Margulis, Morgenstern]
\label{thm:ramanujan}
For any odd prime power $p$, there exist infinitely many (bipartite) $(p+1)$-regular Ramanujan graphs $X^{p,q}$. The graph $X^{p,q}$ is a Cayley graph on the group $\textrm{PGL}(2,q)$, so has order $q(q^2-1)$. Moreover, its girth satisfies
$$g(X^{p,q}) \geq \frac{4\log q}{\log p} - \frac{\log 4}{\log p}.$$
\end{theorem}

It is in place to remark that recently, Marcus, Spielman and Srivastava \cite{mss} proved the existence of infinitely many $d$-regular Ramanujan graphs for {\em every} $d \geq 3$. They did this by proving a weakening of a conjecture of Bilu and Linial \cite{b-l} on 2-lifts of Ramanujan graphs, namely, that every $d$-regular Ramanujan graph has a 2-lift whose second-largest eigenvalue is at most $2\sqrt{d-1}$. Their proof uses a beautiful new technique for demonstrating the existence of combinatorial objects, which they call the `method of interlacing polynomials'. (Even more spectacularly, they use this method to prove the Kadison-Singer conjecture, in \cite{ks}.) Being non-constructive, however, their proof does not imply good bounds for the girth problem.

We are able to improve upon the lower bound in Corollary \ref{corr:lowerbound} when $r=3$ and $d=2$, using the following explicit construction, based upon the Ramanujan graphs of Theorem \ref{thm:ramanujan}. Let $G$ be an $n$-vertex, $3$-regular graph of girth $g$. Take any drawing of $G$ in the plane with straight-line edges, and for each edge $e \in E(G)$, let $m(e)$ be its midpoint. Let $H$ be the 3-uniform hypergraph with
\begin{align*}
V(H) &= \{m(e):\ e \in E(G)\},\\
E(H) &= \{\{m(e_1),m(e_2),m(e_3)\}:\ e_1,e_2,e_3 \textrm{ are incident to a common vertex of }G\}.\end{align*}

Then the hypergraph $H$ is $2$-regular, and also has girth $g$. Taking $G = X^{2,q}$ (the Ramanujan graph of Theorem \ref{thm:ramanujan}) yields a 3-uniform, 2-regular hypergraph $H$ with 
\begin{align*} g(H) & = g(X^{2,q})\\
& \geq \frac{4\log q}{\log 2} - 2\\
& \geq \frac{4}{3} \frac{\log n}{\log 2}-2\end{align*}
improving upon the bound in Corollary \ref{corr:lowerbound} by a factor of $\tfrac{4}{3} - o(1)$.

The following explicit construction, also based on the Ramanujan graphs of Theorem \ref{thm:ramanujan}, provides $r$-uniform, $d$-regular hypergraphs of girth approximately $2/3$ of  the bound in Corollary \ref{corr:lowerbound}, whenever $d$ is a multiple of $r$. (We thank an anonymous referee of an earlier version of this paper, for pointing out this construction.) 

Suppose $d=rs$ for some $s \in \mathbb{N}$. Let $G$ be a $2(r-1)s$-regular, $n$ by $n$ bipartite graph, with vertex-classes $X$ and $Y$, and girth $g$. Then the edge-set of $G$ may be partitioned into $(r-1)$-edge stars in such a way that each vertex of $G$ is in exactly $rs$ of the stars. (Indeed, by Hall's theorem, we may partition the edge-set of $G$ into $2(r-1)s$ perfect matchings. First, choose $r-1$ of these matchings, and group the edges of these matchings into $n$ $(r-1)$-edge stars with centres in $X$. Now choose $r-1$ of the remaining matchings, and group their edges into $n$ $(r-1)$-edge stars with centres in $Y$. Repeat this process $s$ times to produce the desired partition of $E(G)$ into stars.)

Let $H$ be the $r$-uniform hypergraph whose vertex-set is $X \cup Y$, and whose edge-set is the collection of vertex-sets of these stars; then $H$ is $(rs)$-regular, and has girth at least $g/2$.

If $2(r-1)s-1$ is a prime power, the bipartite Ramanujan graph $X^{p,q}$ (with $p = 2(r-1)s-1$) can be used to supply the graph $G$.
This yields a linear, $r$-uniform, $(rs)$-regular hypergraph with girth $g(H)$ satisfying
\begin{align*}g(H)& \geq \frac{1}{2}\left(\frac{4\log q}{\log (2rs-2s-1)} - \frac{\log 4}{\log (2rs-2s-1)}\right)\\
& \geq \frac{1}{2}\left(\frac{4}{3}\frac{\log n}{\log (2rs-2s-1)} - \frac{\log 4}{\log (2rs-2s-1)}\right)\\
& = \frac{2}{3}\frac{\log n}{\log (2d-2d/r-1)} - \frac{\log 2}{\log (2d-2d/r-1)},\end{align*}
where $d=rs$.

Unfortunately, this lower bound is asymptotically worse than that given by Corollary \ref{corr:lowerbound}, for all values of $r$ and $d$.

\begin{comment}

\section{An explicit construction}

In this section, we give an explicit construction of $r$-uniform, $d$-regular, linear hypergraphs of large girth, for $d$ a multiple of $r$. This construction will beat the lower bound in Corollary \ref{corr:lowerbound} for only two pairs $(r,d)$; its main advantage is that it is explicit.

Suppose $d=rs$. Suppose $G$ is a Cayley graph on a group $\Gamma$, with symmetric generating set of size $2(r-1)s$ and with girth $g > 2(r-1)$. Suppose its generators are $x_1,x_2,\ldots,x_{(r-1)s}$ and their inverses. Let
$$a^{(j)}_{i} = x_{(j-1)r+i}\quad (i \in [r-1],\ j \in [s]).$$
We construct a suitable hypergraph from $G$, as follows. Let $H$ be the hypergraph with vertex-set $\Gamma$ and edge-set
$$\{\{g, ga^{(j)}_1, ga^{(j)}_1a^{(j)}_2,\ldots, ga^{(j)}_1 a^{(j)}_2 \ldots a^{(j)}_{r-1}\}:\ j \in [s],\ g \in \Gamma\}.$$

We need two more definitions. If $S$ is a set of symbols, a {\em word} of $S$ is a string of the form
$$s_1^{a_1}s_2^{a_2} \ldots s_L^{a_L}$$
where $s_1,\ldots,s_L \in S$. Such a word is said to be {\em cyclically irreducible} if $s_i \neq s_{i+1}$ for all $i \in [L]$, where we define $s_{L+1} := s_1$.

 sequence of elements of a group, we say that the word $g_1g_2\ldots g_N$ is {\em irreducible} if $g_i \neq g_{i+1}^{-1}$ for all $i \in [N]$ (where we define $g_{N+1} := g_1$). Otherwise, we say that it is {\em reducible}. Note that the girth of any Cayley graph is the length of the shortest irreducible word whose letters are generators of the graph, and which evaluates to the identity element of the group.

To analyse our hypergraph $H$, we will need the following.

\begin{claim}
If
$$\{g, ga^{(j)}_1, ga^{(j)}_1a^{(j)}_2,\ldots, ga^{(j)}_1 a^{(j)}_2 \ldots a^{(j)}_{r-1}\} = \{g',g'a^{(k)}_1, g'a^{(k)}_1a^{(k)}_2,\ldots, g'a^{(k)}_1 a^{(k)}_2 \ldots a^{(k)}_{r-1}\},$$
then $j=k$ and $g=g'$. 
\end{claim}
\begin{proof}
Suppose for a contradiction that
$$\{g, ga^{(j)}_1, ga^{(j)}_1a^{(j)}_2,\ldots, ga^{(j)}_1 a^{(j)}_2 \ldots a^{(j)}_{r-1}\} = \{g',g'a^{(k)}_1, g'a^{(k)}_1a^{(k)}_2,\ldots, g'a^{(k)}_1 a^{(k)}_2 \ldots a^{(k)}_{r-1}\}$$
for some $j \neq k$ and some $g,g' \in \Gamma$. For brevity, let us write $a^{(j)}_i = a_i$ and $a^{(k)}_i = b_i$ for all $i$. Then we have
$$\{g, ga_1, ga_1a_2,\ldots, ga_1 a_2 \ldots a_{r-1}\} = \{g',g'b_1, g'b_1b_2,\ldots, g'b_1 b_2 \ldots b_{r-1}\}.$$
Then $g = g'b_1 b_2 \ldots b_l$ for some $l \in \{0,1,\ldots,r-1\}$, and $ga_1 = g' b_1 b_2 \ldots b_m$ for some $m \in \{0,1,\ldots,r-1\}$, where $m \neq l$. Therefore,
$$a_1 = g^{-1}(ga_1) = \left\{\begin{array}{ll} b_{l+1} b_{l+2} \ldots b_m & \textrm{if } l < m;\\ b_{l}^{-1} b_{l-1}^{-1} \ldots b_{m+1}^{-1} & \textrm{if } l > m.\end{array} \right.$$
In either case, we obtain an irreducible word in the generators of $\Gamma$, which has length at most $r$, and evaluates to the identity element of $\Gamma$. This contradicts our assumption that the girth of $G$ is greater than $2(r-1)$.

Assume now that $j=k$. Suppose for a contradiction that
$$\{g, ga^{(j)}_1, ga^{(j)}_1a^{(j)}_2,\ldots, ga^{(j)}_1 a^{(j)}_2 \ldots a^{(j)}_{r-1}\} = \{g',g'a^{(j)}_1, g'a^{(j)}_1a^{(j)}_2,\ldots, g'a^{(j)}_1 a^{(j)}_2 \ldots a^{(j)}_{r-1}\},$$
where $g \neq g'$. As before, let us write $a^{(j)}_i = a_i$ for all $i$. Then we have
$$\{g, ga_1, ga_1a_2,\ldots, ga_1 a_2 \ldots a_{r-1}\} = \{g',g'a_1, g'a_1a_2,\ldots, g'a_1 a_2 \ldots a_{r-1}\}.$$
Hence, we must have $g' = ga_1 a_2 \ldots a_l$ for some $l \in [r-1]$, and $g = g'a_1 a_2 \ldots a_m$ for some $m \in [r-1]$. But then
$$g' = ga_1 a_2 \ldots a_l = (g'a_1 a_2 \ldots a_m)a_1 a_2 \ldots a_l,$$
so
$$a_1 a_2 \ldots a_m a_1 a_2 \ldots a_l = \textrm{Id},$$
and we have an irreducible word in the generators of $G$, which has length at most $2(r-1)$, and evaluates to the identity, contradicting our assumption that $G$ has girth greater than $2(r-1)$.
\end{proof} 

The above claim implies that $H$ is $(rs)$-regular. Indeed, each $h \in \Gamma$ is contained only in the edges
$$\{g, ga^{(j)}_1,ga^{(j)}_1a^{(j)}_2, \ldots, ga^{(j)}_1 a^{(j)}_2 \ldots a^{(j)}_{r-1}\}$$
obtained by choosing $g = h(a^{(j)}_l)^{-1} (a^{(j)}_{l-1})^{-1} \ldots (a^{(j)}_1)^{-1}$ (for some $j \in [s]$ and some $l \in \{0,1,\ldots,r-1\}$), and each of these choices produces a different edge, by the above claim.

We now make the following.

\begin{claim}
The hypergraph $H$ is linear.
\end{claim}
\begin{proof}
Suppose that two distinct vertices $v_1$ and $v_2$ both lie in the edge $e$. We will show that they cannot both lie in any other edge. Suppose
$$e = \{g,ga^{(j)}_1,ga^{(j)}_1a^{(j)}_2,\ldots,ga^{(j)}_1a^{(j)}_2\ldots a^{(j)}_{r-1}\}.$$
For brevity, let us write $a^{(j)}_i = a_i$ for all $i$, so that
$$e =  \{g,ga_1,ga_1a_2,\ldots,ga_1a_2\ldots a_{r-1}\}.$$
Then we must have $v_1 = ga_1a_2\ldots a_l$ and $v_2 = ga_1 a_2 \ldots a_m$ for some distinct $l,m \in \{0,1,2,\ldots,r-1\}$. Therefore,
$$v_1^{-1}v_2 = \left\{\begin{array}{ll} a_{l+1}a_{l+2}\ldots a_m & \textrm{if }l < m;\\ a_l^{-1} a_{l-1}^{-1} \ldots a_{m+1}^{-1} & \textrm{if } l > m.\end{array}\right.$$
Suppose for a contradiction that $v_1$ and $v_2$ also both lie in some other edge $f$. Suppose firstly that
$$f=\{g',g'a^{(k)}_1, g'a^{(k)}_1a^{(k)}_2,\ldots, g'a^{(k)}_1 a^{(k)}_2 \ldots a^{(k)}_{r-1}\}$$
for some $k \neq j$. For brevity, let us write $a^{(k)}_i = b_i$ for all $i$, so that
$$f = \{g', g'b_1, g'b_1b_2,\ldots,g'b_1 b_2 \ldots b_{r-1}\}.$$
Then, using the same argument as above, $v_1^{-1}v_2$ is equal to some word of length at most $r-1$ with letters from $\{b_1,b_2,\ldots,b_{r-1}\}$ (or from $\{b_1^{-1},b_2^{-1},\ldots,b_{r-1}^{-1}\}$). Equating the two expressions for $v_1 v_2^{-1}$ yields an irreducible word of length at most $2(r-1)$ which evaluates to the identity element of $\Gamma$. This contradicts our assumption that the girth of $G$ is greater than $2(r-1)$.

Suppose secondly that $k=j$. Then, using $f$ instead of $e$, we may write $v_1 v_2^{-1}$ as a word of length at most $r-1$ with letters from $\{a_1,a_2,\ldots,a_{r-1}\}$ (or from $\{a_1^{-1},a_2^{-1},\ldots,a_{r-1}^{-1}\}$. The word obtained using $f$ must be the same as that obtained using $e$, otherwise we would again obtain an irreducible word of length at most $2(r-1)$ which evaluates to the identity element of $\Gamma$. It follows that we must have $v_1 = ga_1 a_2 \ldots a_l = g'a_1 a_2 \ldots a_l$ for some $l$, and therefore $g=g'$, so $e=f$. This completes the proof of the claim.
\end{proof}

Finally, we make the following.

\begin{claim}
The hypergraph $H$ has girth at least $g/(r-1)$.
\end{claim}
\begin{proof}
Suppose $H$ has a cycle of length $t$. Let $(e_1,e_2,\ldots,e_t)$ be any cyclic ordering of its edges. Since $H$ is linear, we have $|e_i \cap e_{i+1}|=1$ for all $i \in [t]$ (where we define $e_{t+1}:=e_1$); let $e_i \cap e_{i+1}=\{w_i\}$ for each $i \in [t]$. Fix $i \in [t]$. Note that $w_i,w_{i+1} \in e_{i+1}$. Suppose
$$e_{i+1} = \{g,ga^{(j)}_1,ga^{(j)}_1a^{(j)}_2,\ldots,ga^{(j)}_1a^{(j)}_2\ldots a^{(j)}_{r-1}\}.$$
Then we have $w_i = ga^{(j)}_1a^{(j)}_2\ldots a^{(j)}_{l}$ and $w_{i+1} = ga^{(j)}_1a^{(j)}_2\ldots a^{(j)}_{m}$ for some distinct $l,m \in \{0,1,\ldots,r-1\}$ (depending on $i$). Therefore,
\begin{equation}\label{eq:expression}w_i^{-1}w_{i+1} = \left\{\begin{array}{ll} a^{(j)}_{l+1}a^{(j)}_{l+2}\ldots a^{(j)}_m & \textrm{if }l < m;\\ (a^{(j)}_l)^{-1} (a^{(j)}_{l-1})^{-1} \ldots (a^{(j)}_{m+1})^{-1} & \textrm{if } l > m.\end{array}\right.\end{equation}
Clearly, we have
\begin{equation}\label{eq:word} (w_1^{-1}w_2)(w_2^{-1}w_3)(w_3^{-1} w_4) \ldots (w_{l-1}^{-1}w_l)(w_l^{-1}w_1) = \textrm{Id}.\end{equation}
Substituting the expressions in (\ref{eq:expression}) for each factor $(w_i^{-1}w_{i+1})$ in the left-hand side of (\ref{eq:word}), we obtain a word of length at most $(r-1)l$ which evaluates to the identity element of $\Gamma$. We must check that this word is irreducible. Suppose for a contradiction that it is reducible. Then there exists $i \in [t]$ such that the last letter of the expression for $(w_i^{-1}w_{i+1})$ is the inverse of the first letter of the expression for $(w_{i+1}^{-1}w_{i+2})$. Therefore, we must have
$$e_{i+1} = \{g,ga^{(j)}_1,ga^{(j)}_1a^{(j)}_2,\ldots,ga^{(j)}_1a^{(j)}_2\ldots a^{(j)}_{r-1}\}$$
and
$$e_{i+2} = \{g',g'a^{(j)}_1,g'a^{(j)}_1a^{(j)}_2,\ldots,g'a^{(j)}_1a^{(j)}_2\ldots a^{(j)}_{r-1}\}$$
for some $j \in [s]$ and some (distinct) $g,g' \in \Gamma$. For brevity, let us write $a^{(j)}_i=a_i$ for all $i$; then
$$e_{i+1} = \{g,ga_1,ga_1a_2,\ldots,ga_1a_2\ldots a_{r-1}\}$$
and
$$e_{i+2} = \{g',g'a_1,g'a_1a_2,\ldots,g'a_1a_2\ldots a_{r-1}\}.$$

We have $w_i = ga_1 a_2 \ldots a_l$ and $w_{i+1} = ga_1 a_2 \ldots a_{m}$ for some distinct $l,m \in \{0,1,\ldots,r-1\}$, and we also have $w_{i+1} = g'a_1 a_2 \ldots a_{l'}$ and $w_{i+2} = g'a_1 a_2 \ldots a_{m'}$ for some distinct $l',m' \in \{0,1,\ldots,r-1\}$. Therefore,
$$w_i^{-1}w_{i+1} = \left\{\begin{array}{ll} a_{l+1}a_{l+2}\ldots a_m & \textrm{if }l < m;\\ a_l^{-1} a_{l-1}^{-1} \ldots a_{m+1}^{-1} & \textrm{if } l > m,\end{array}\right.$$
and
$$w_{i+1}^{-1}w_{i+2} = \left\{\begin{array}{ll} a_{l'+1}a_{l'+2}\ldots a_{m'} & \textrm{if }l' < m';\\ a_{l'}^{-1} a_{l'-1}^{-1} \ldots a_{m'+1}^{-1} & \textrm{if } l' > m'.\end{array}\right.$$

Since the last letter of the above expression for $w_i^{-1}w_{i+1}$ is the inverse of the first letter of the above expression for $w_{i+1}^{-1}w_{i+2}$, we must have $m=l'$, and therefore
$$ga_1 a_2 \ldots a_{m} = w_{i+1} = g'a_1a_2 \ldots a_{m},$$
so $g=g'$, so $e_{i+1}=e_{i+2}$, a contradiction.

Hence, the word in (\ref{eq:word}) is irreducible, as claimed. This word has length at most $(r-1)l$, and evaluates to the identity element of $\Gamma$. Since $G$ has girth $g$, we must have $l \geq g/(r-1)$. Hence, $H$ has girth at least $g/(r-1)$, proving the claim.
\end{proof}

The construction of our hypergraph $H$ requires an $((r-1)s)$-regular Cayley graph of high girth. When $2(r-1)s-1$ is a prime power, the bipartite Ramanujan graphs constructed by Lubotzsky, Phillips and Sarnak \cite{lps}, Margulis \cite{margulis}, and Morgenstern \cite{morgenstern} can be used.

\begin{theorem}[Lubotzsky-Phillips-Sarnak, Margulis, Morgenstern]
For any odd prime power $p$, there exist infinitely many (bipartite) $(p+1)$-regular Ramanujan graphs $X^{p,q}$. The graph $X^{p,q}$ is a Cayley graph on the group $\textrm{PGL}(2,q)$, so has order $q(q^2-1)$. Its girth satisfies
$$g(X^{p,q}) \geq \frac{4\log q}{\log p} - \frac{\log 4}{\log p}.$$
\end{theorem}

Hence, when $2rs-2s-1$ is a prime power, then if we use $X^{p,q}$ as our graph $G$, our construction yields a linear, $r$-uniform, $(rs)$-regular hypergraph with girth $g(H)$ satisfying
\begin{align*}g(H)& \geq \frac{1}{r-1}\left(\frac{4\log q}{\log (2rs-2s-1)} - \frac{\log 4}{\log (2rs-2s-1)}\right)\\
& \geq \frac{1}{r-1}\left(\frac{4}{3}\frac{\log n}{\log (2rs-2s-1)} - \frac{\log 4}{\log (2rs-2s-1)}\right)\\
& = \frac{1}{r-1}\left(\frac{4}{3}\frac{\log n}{\log (2d-2d/r-1)} - \frac{\log 4}{\log (2d-2d/r-1)}\right),\end{align*}
where $d=rs$.

This yields the following.

\begin{theorem}
Let $r \geq 3$, and let $d$ be a multiple of $r$ such that $2d-2d/r-1$ is a prime power. Then for infinitely many $n$, there exists an $r$-uniform, $d$-regular, linear hypergraph with $n$ vertices and girth at least
$$ \frac{1}{r-1}\left(\frac{4}{3}\frac{\log n}{\log (2d-2d/r-1)} - \frac{\log 4}{\log (2d-2d/r-1)}\right).$$
\end{theorem}

\end{comment}

\section{Random `Cayley' hypergraphs}
 In this section, we give a construction of random `Cayley' hypergraphs on the symmetric group $S_n$, which have girth $\Omega(\sqrt{\log |S_n|})$ with high probability. This is much higher than the girth of a random regular hypergraph on the same number of vertices (which, by (\ref{eq:randomreg}), has girth at most $C(\epsilon)$ with probability at least $1-\epsilon$ for any $\epsilon >0$, where $C(\epsilon)$ is a constant depending on $\epsilon$ alone), though it is still short of the optimal $\Theta(\log |V(H)|)$ in Corollary \ref{corr:lowerbound}. The situation is analogous to the graph case, where random $d$-regular Cayley graphs on appropriate groups have much higher girth than random $d$-regular graphs of the same order (due to the dependency between cycles at different vertices of a Cayley graph).
 
First, we need some more definitions. If $S$ is a set of symbols, a {\em word in} $S$ is a string of the form
$$s_1^{a_1}s_2^{a_2} \ldots s_l^{a_l}$$
where $s_1,\ldots,s_l \in S$ and $a_1,\ldots,a_l \in \mathbb{Z} \setminus \{0\}$. Such a word is said to be {\em cyclically irreducible} if $s_i \neq s_{i+1}$ for all $i \in [l]$, where we define $s_{l+1} := s_1$. Its {\em length} is $\sum_{i=1}^{l}|a_i|$.

\begin{theorem}
\label{thm:sym}
Let $r$ and $n$ be positive integers with $r \geq 3$ and $r|n$. Let $X(n,r)$ be the set of permutations in $S_n$ that consist of $\frac nr$ disjoint $r$-cycles. Choose $d$ permutations $\tau_1,\tau_2,\ldots,\tau_d$ uniformly at random and independently (with replacement) from $X(n,r)$, and let $H$ be the random hypergraph with vertex-set $S_n$ and edge-set
$$\{\{\sigma,\sigma \tau_i, \sigma \tau_i^2,\ldots,\sigma \tau_i^{r-1}\}:\ \sigma \in S_n,\ i \in [d]\}.$$
Then with high probability, $H$ is a linear, $r$-uniform, $d$-regular hypergraph with girth at least
$$c_0 \sqrt{\frac{n \log n}{r(r-1)(\log(d-1)+\log(r-1))}},$$
for any absolute constant $c_0$ such that $0 < c_0 < 1/2$.
\end{theorem}

\begin{remark}
Here, `with high probability' means `with probability tending to 1 as $n \to \infty$'.
\end{remark}
\begin{proof}
Note that the edges of the form
$$\{\sigma,\sigma \tau_i, \sigma \tau_i^2,\ldots,\sigma \tau_i^{r-1}\}\ (\sigma \in S_n)$$
are simply the left cosets of the cyclic group $\{\textrm{Id},\tau_i,\tau_i^2,\ldots,\tau_i^{r-1}\}$ in $S_n$, so they form a partition of $S_n$. We need two straightforward claims.
\begin{claim}
\label{claim:condition}
With high probability, the following condition holds.
\begin{equation}\label{eq:cond} \tau_1,\ldots,\tau_d \textrm{ satisfy}\quad \tau_i^{k} \neq \tau_j^{l}\quad \textrm{for all distinct }i,j \in [d] \textrm{ and all }k,l \in [r-1].\end{equation}
\end{claim}
\begin{proof}[Proof of claim:]
Let us fix $i,j \in [d]$ with $i <j$, and fix $k,l \in [r-1]$. We shall bound the probability that $\tau_j^{l} = \tau_i^{k}$. We regard $\tau_i$ as fixed, and allow $\tau_j$ to vary. Since $\tau_i$ is a product of $n/r$ disjoint $r$-cycles, $\tau_i^k$ is a product of $n/s$ disjoint $s$-cycles, for some integer $s \ge 2$ that is a divisor of $r$. The set $X(n,s)$ of permutations which consist of $n/s$ disjoint $s$-cycles has cardinality
$$\frac{n!}{(n/s)! s^{n/s}} \geq \frac{n!}{(n/2)!2^{n/2}}$$
(provided $n \geq 4$). Notice that $\tau_j^{l}$ is uniformly distributed over $X(n,s')$, for some $s'$ that depends only on $r$ and $l$. Therefore,
$$\textrm{Prob}\{\tau_i^k = \tau_j^l\} \leq \frac{(n/2)!2^{n/2}}{n!}.$$
By the union bound,
\begin{align*}\textrm{Prob}\{\tau_i^k = \tau_j^l \textrm{ for some }i \neq j \textrm{ and some }k,l \in [r-1]\} & \leq (r-1)^2{d \choose 2} \frac{(n/2)!2^{n/2}}{n!}\\
& \to 0 \quad \textrm{as }n \to \infty,\end{align*}
proving the claim.
\end{proof}

\begin{claim}
\label{claim:linear}
If condition (\ref{eq:cond}) holds, then for all $i \neq j$ and all $\sigma, \pi \in S_n$, the two cosets
$$\{\sigma, \sigma \tau_i, \sigma \tau_i^2,\ldots, \sigma \tau_i^{r-1}\} \mbox{~~and~~} \{\pi, \pi \tau_j, \pi \tau_j^2, \ldots, \pi \tau_j^{r-1}\}$$
have at most one element in common.
\end{claim}
\begin{proof}[Proof of claim:]
Suppose for a contradiction that there are two distinct vertices $v_1,v_2$ with
$$v_1,v_2 \in \{\sigma, \sigma \tau_i, \sigma \tau_i^2,\ldots, \sigma \tau_i^{r-1}\} \cap \{\pi, \pi \tau_j, \pi \tau_j^2, \ldots, \pi \tau_j^{r-1}\}.$$
Then $v_1 = \sigma \tau_i^l = \pi \tau_j^{m}$ and $v_2 = \sigma \tau_i^{l'} = \pi \tau_j^{m'}$, where $l,m,l',m' \in \{0,1,\ldots,r-1\}$ with $l' \neq l$ and $m' \neq m$. Therefore,
$$v_1^{-1}v_2 = \tau_i^{l'-l} = \tau_j^{m'-m},$$
contradicting condition (\ref{eq:cond}).
\end{proof}

Claim \ref{claim:linear} implies that $H$ is a linear hypergraph, provided condition (\ref{eq:cond}) is satisfied. Moreover, $H$ is $d$-regular: every $\sigma \in S_n$ is contained in the edges (cosets)
$$(\{\sigma, \sigma \tau_i,\sigma \tau_i^2,\ldots,\sigma \tau_i^{r-1}\}: i \in [d]),$$
and these $d$ edges are distinct provided condition (\ref{eq:cond}) is satisfied.

Finally, we make the following.
\begin{claim}
\label{claim:girth}
With high probability, $H$ has girth at least
$$c_0 \sqrt{\frac{n \log n}{r(r-1)(\log(d-1)+\log(r-1))}},$$
where $c_0$ is any absolute constant such that $0 < c_0 < 1/2$.
\end{claim}
\begin{proof}[Proof of claim:]
We may assume that condition (\ref{eq:cond}) holds, so that $H$ is a linear, $d$-regular hypergraph. Let $C$ be a cycle in $H$ of minimum length, and let $(e_1,\ldots,e_l)$ be any cyclic ordering of its edges. Then we have $|e_i \cap e_{i+1}|=1$ for all $i \in [l]$ (where we define $e_{l+1} := e_1$), and by minimality, we have $e_i \cap e_j = \emptyset$ whenever $|i-j| > 1$. Let $e_i \cap e_{i+1} = \{w_i\}$ for each $i \in [l]$. Suppose that $e_i$ is an edge of the form
$$\{\sigma, \sigma \tau_{j_i}, \sigma \tau_{j_i}^2,\ldots, \sigma \tau_{j_i}^{r-1}\}$$
for each $i \in [l]$. Since $e_i \cap e_{i+1} \neq \emptyset$ for each $i \in [l]$, we must have $j_i \neq j_{i+1}$ for all $i \in [l]$ (where we define $j_{l+1}:=j_1$). For each $i \in [l]$, we have $w_i,w_{i+1} \in e_{i+1}$, so $w_i^{-1}w_{i+1} = \tau_{j_{i+1}}^{m_i}$ for some $m_i \in [r-1]$. Therefore,
\begin{equation}\label{eq:irreducibleword}\textrm{Id} = (w_1^{-1}w_2)(w_2^{-1}w_3)\ldots (w_{l-1}^{-1}w_l)(w_{l}^{-1}w_1) = \tau_{j_2}^{m_1} \tau_{j_3}^{m_2} \ldots \tau_{j_l}^{m_{l-1}} \tau_{j_{1}}^{m_l}.\end{equation}
Since $j_i \neq j_{i+1}$ for all $i \in [l]$, the word on the right-hand side of (\ref{eq:irreducibleword}) is cyclically irreducible. We therefore have a cyclically irreducible word in the symbols $\{\tau_j:\ j \in [d]\}$ with length $L:=\sum_{j=1}^{l}m_i \leq (r-1)l$, which evaluates to the identity permutation. We must show that the probability of this tends to zero as $n \to \infty$, for an appropriate choice of $l$. We use an argument similar to that of \cite{ghssv}, where it is proved that a random $d$-regular Cayley graph on $S_n$ has girth at least $\Omega(\sqrt{\log_{d-1} (n!)})$.

Let $W$ be a cyclically irreducible word in the $\tau_j$'s, with length $L$. We must bound the probability that $W$ fixes every element of $[n]$. Suppose
$$W = \tau_{j(1)} \tau_{j(2)}\ldots \tau_{j(L)}.$$
Let $x_0 \in [n]$, and define $x_{i} = \tau_{j(i)}(x_{i-1})$ for each $i \in [L]$, producing a sequence of values $x_0,x_1,x_2,\ldots,x_L \in [n]$; then $W(x_0) = x_L$. We shall bound the probability that $x_L = x_0$. Let us work our way along the sequence, exposing the $r$-cycles of the permutations $\tau_1,\ldots,\tau_d$ only as we need them, so that at stage $i$, the $r$-cycle of $\tau_{j(i)}$ containing the number $x_{i-1}$ is exposed (if it has not already been exposed). If $x_L = x_0$, then (as $j(L) \neq j(1)$), there has to be a first time the sequence returns to $x_0$ via a permutation $\tau \neq \tau_{j(1)}$. Hence, at some stage, we must have exposed an $r$-cycle of $\tau$ containing $x_0$. The probability that, at a stage $i$ where $j(i) \neq j(1)$, we expose an $r$-cycle of $\tau_{j(i)}$ containing $x_0$, is at most
$$\frac{r}{n-(i-2)r} \leq \frac{r}{n-(L-2)r},$$
since a total of at most $i-2$ $r$-cycles of $\tau$ have already been exposed, and the next $r$-cycle exposed is equally likely to be any $r$-element subset of the remaining $n-(i-2)r$ numbers. There are at most $L$ choices for the stage $i$, and therefore
$$\textrm{Prob}\{W(x_0)=x_0\} \leq L \frac{r}{n-(L-2)r}.$$

Suppose we have already verified that $W$ fixes $y_1,y_2,\ldots,y_{m-1}$, by exposing the necessary $r$-cycles. Then we have exposed at most $(m-1)L$ $r$-cycles. As long as $(m-1)Lr < n$, we can choose a number $y_m \in [n]$ such that none of the previously exposed $r$-cycles contains $y_m$. Repeating the above argument yields an upper bound of
$$\frac{Lr}{n-mLr}$$
on the probability that $W$ fixes $y_m$, even when conditioning on the $(m-1)L$ previously exposed $r$-cycles. Therefore,
$$\textrm{Prob}\{W = \textrm{Id}\} \leq \left(\frac{Lr}{n-mLr}\right)^m,$$
as long as $mLr < n$. Substituting $m = \lceil n/(2Lr) \rceil$ yields the bound
$$\textrm{Prob}\{W = \textrm{Id}\} \leq \left(\frac{2Lr}{n}\right)^{n/(2Lr)}.$$
The number of choices for the word on the right-hand side of (\ref{eq:irreducibleword}) is at most $(d-1)^{l} (r-1)^l$. (By taking a cyclic shift if necessary, we may assume that $j_2 \neq d$, so there are at most $d-1$ choices for $j_2$, and at most $d-1$ choices for all subsequent $j_i$; there are clearly at most $r-1$ choices for each $m_i$.) Hence, the probability that there exists such a word which evaluates to the identity permutation is at most
$$(d-1)^{l} (r-1)^l \left(\frac{2r(r-1)l}{n}\right)^{n/(2r(r-1)l)}.$$
To bound the probability that $H$ has a cycle of length less than $g$, we need only sum the above expression over all $l < g$:
\begin{align*} \textrm{Prob}\{\textrm{girth}(H) < g\} & \leq \sum_{l=3}^{g-1} (d-1)^{l} (r-1)^l \left(\frac{2r(r-1)l}{n}\right)^{n/(2r(r-1)l)}\\
& < (d-1)^{g} (r-1)^g \left(\frac{2r(r-1)g}{n}\right)^{n/(2r(r-1)g)}.\end{align*}
In order for the right-hand side to tend to zero as $n \to \infty$, we must choose
$$g = c_0 \sqrt{\frac{n \log n}{r(r-1)(\log(d-1)+\log(r-1))}}$$
for some constant $c_0 < 1/2$; we then have
$$\textrm{Prob}\{\textrm{girth}(H) < g\} \leq \exp\left(-\Omega\left(\frac{1}{r}\sqrt{(\log(d-1)+\log(r-1))(n \log n)}\right)\right).$$
This completes the proof of Claim \ref{claim:girth}, and thus proves Theorem \ref{thm:sym}.
\end{proof}
\end{proof}


\section{Conclusion and open problems}\label{sec:open}
Our best (general) upper and lower bounds on the function $g_{r,d}(n)$ differ approximately by a factor of 2:
$$(1+o(1))\frac{\log n}{\log(d-1)+\log(r-1)} \leq g_{r,d}(n) \leq (2+o(1))\frac{\log n}{\log(r-1)+\log(d-1)}.$$
It would be of interest to narrow the gap, possibly by means of an explicit algebraic construction {\em \`a la} Ramanujan graphs.

In \cite{ghssv}, Gamburd, Hoory, Shahshahani, Shalev and Vir\'ag conjecture that with high probability, a random $d$-regular Cayley graph on $S_n$ has girth at least $\Omega(\log |S_n|)$, as opposed to the $\Omega(\sqrt{\log |S_n|})$ which they prove. We believe that the random hypergraph of Theorem \ref{thm:sym} also has girth $\Omega(\log |S_n|)$.

In this paper, we considered a very simple and purely combinatorial notion of girth in hypergraphs, but other notions appear in the literature, for example using the language of simplicial topology, such as in \cite{lub:mesh, goff}. A different combinatorial definition was introduced by Erd\H{o}s in \cite{erdos:sts}. Define the $(-2)$-girth of a $3$-uniform hypergraph as the smallest integer $g \geq 4$ such that there is a set of $g$ vertices spanning at least $g-2$ edges. Erd\H{o}s conjectured in \cite{erdos:sts} that there exist Steiner Triple Systems with arbitrarily high $(-2)$-girth; this question remains wide open (see for example \cite{beezer}), and seems very hard.  In view of this, we raise the following.

\begin{question}
\label{question:quad}
Is there a constant $c>0$ such that there exist $n$-vertex $3$-uniform hypergraphs with $cn^2$ edges and arbitrarily high $(-2)$-girth?
\end{question}

Note that Erd\H{o}s' conjecture on Steiner Triple Systems, if true, would imply a positive answer for every $c<\frac 16$. This is clearly tight, since an $n$-vertex, 3-uniform hypergraph with at least $n^2/6$ edges cannot be linear,\footnote{If $H$ is a linear, $n$-vertex, 3-uniform hypergraph, then any pair of vertices is contained in at most one edge of $H$, so double-counting the number of times a pair of vertices in contained in an edge of $H$, we obtain $3e(H) \leq {n \choose 2}$.} and therefore has $(-2)$-girth 4. 
%It is easy to see that any $n$-vertex, 3-uniform hypergraph with at least $n^2/3$ edges %must contain a set of $g$ vertices spanning at least $g-2$ edges for {\em every} $g \in %\{4,5,\ldots,n\}$, so necessarily $c < 1/3$.

We turn briefly to some variants of Erd\H{o}s' definition. The celebrated $(6,3)$-theorem of Ruzsa and Szemer\'edi~\cite{ruzs:sz} states that if $H$ is an $n$-vertex, $3$-uniform hypergraph in which no 6 vertices span 3 or more edges, then $H$ has $o(n^2)$ edges. Therefore, if we define the $(-3)$-girth of a 3-uniform hypergraph to be the smallest integer $g \geq 6$ such that there exists a set of $g$ vertices spanning at least $g-3$ edges,\footnote{The condition $g \geq 6$ is necessary to avoid triviality: if we replaced it with $g \geq 5$, then a 3-uniform hypergraph would have $(-3)$-girth 5 unless it consisted of isolated edges.} then an $n$-vertex, 3-uniform hypergraph with $(-3)$-girth at least 7 has $o(n^2)$ edges. Hence, the analogue of Question \ref{question:quad} for $(-3)$-girth has a negative answer. On the other hand, if we define the $(-1)$-girth of a 3-uniform hypergraph to be the smallest integer $g$ such that there exists a set of $g$ vertices spanning at least $g-1$ edges, it can be shown that the maximum number of edges in an $n$-vertex, 3-uniform hypergraph with $(-1)$-girth at least $g$, is $n^{2+\Theta(1/g)}.$ \begin{comment} We will discuss these and related problems more fully in a forthcoming note \cite{note}. \end{comment}

\section{Acknowledgment}
N. L. wishes to thank Shlomo Hoory for many years of joint pursuit of the girth problem~\cite{hoory}. Quite a few of the ideas of this present paper can be traced back to that joint research effort.

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\end{document}
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