  Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a
set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same
cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all
$r$-subsets of $V$, then $H$ is a complete $r$-uniform hypergraph, denoted by
$K_n^r$, where $n=|V|$. A hypergraph $H'=(V',E')$ is called a subhypergraph of
$H=(V,E)$ if $V'\subseteq V$ and $E'\subseteq E$. A $r$-uniform hypergraph
$H=(V,E)$ is vertex-$k$-maximal if every subhypergraph of $H$ has
vertex-connectivity at most $k$, but for any edge $e\in E(K_n^r)\setminus
E(H)$, $H+e$ contains at least one subhypergraph with vertex-connectivity at
least $k+1$. In this paper, we first prove that for given integers $n,k,r$ with
$k,r\geq2$ and $n\geq k+1$, every vertex-$k$-maximal $r$-uniform hypergraph $H$
of order $n$ satisfies $|E(H)|\geq (^n_r)-(^{n-k}_r)$, and this lower bound is
best possible. Next, we conjecture that for sufficiently large $n$, every
vertex-$k$-maximal $r$-uniform hypergraph $H$ on $n$ vertices satisfies
$|E(H)|\leq(^n_r)-(^{n-k}_r)+(\frac{n}{k}-2)(^k_r)$, where $k,r\geq2$ are
integers. And the conjecture is verified for the case $r>k$.
