  An $\mathcal{F}$-saturated $r$-graph is a maximal $r$-graph not containing
any member of $\mathcal{F}$ as a subgraph. Let $\mathcal{K}_{\ell + 1}^{r}$ be
the collection of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges
such that for some $\left(\ell+1\right)$-set $S$ every pair $\{u, v\} \subset
S$ is covered by an edge in $F$. Our first result shows that for each $\ell
\geq r \geq 2$ every $\mathcal{K}_{\ell+1}^{r}$-saturated $r$-graph on $n$
vertices with $t_{r}(n, \ell) - o(n^{r-1+1/\ell})$ edges contains a complete
$\ell$-partite subgraph on $(1-o(1))n$ vertices, which extends a stability
theorem for $K_{\ell+1}$-saturated graphs given by Popielarz, Sahasrabudhe and
Snyder. We also show that the bound is best possible. Our second result is
motivated by a celebrated theorem of Andr\'{a}sfai, Erd\H{o}s and S\'{o}s which
states that for $\ell \geq 2$ every $K_{\ell+1}$-free graph $G$ on $n$ vertices
with minimum degree $\delta(G) > \frac{3\ell-4}{3\ell-1}n$ is $\ell$-partite.
We give a hypergraph version of it. The \emph{minimum positive co-degree} of an
$r$-graph $\mathcal{H}$, denoted by $\delta_{r-1}^{+}(\mathcal{H})$, is the
maximum $k$ such that if $S$ is an $(r-1)$-set contained in a edge of
$\mathcal{H}$, then $S$ is contained in at least $k$ distinct edges of
$\mathcal{H}$. Let $\ell\ge 3$ be an integer and $\mathcal{H}$ be a
$\mathcal{K}_{\ell+1}^3$-saturated $3$-graph on $n$ vertices. We prove that if
either $\ell \ge 4$ and $\delta_{2}^{+}(\mathcal{H}) >
\frac{3\ell-7}{3\ell-1}n$; or $\ell = 3$ and $\delta_{2}^{+}(\mathcal{H}) >
2n/7$, then $\mathcal{H}$ is $\ell$-partite; and the bound is best possible.
This is the first stability result on minimum positive co-degree for
hypergraphs.
