An F-saturated r-graph is a maximal r-graph not containing
any member of F as a subgraph. Let Kβ+1rβ be
the collection of all r-graphs F with at most (2β+1β) edges
such that for some (β+1)-set S every pair {u,v}βS is covered by an edge in F. Our first result shows that for each ββ₯rβ₯2 every Kβ+1rβ-saturated r-graph on n
vertices with trβ(n,β)βo(nrβ1+1/β) edges contains a complete
β-partite subgraph on (1βo(1))n vertices, which extends a stability
theorem for Kβ+1β-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 ββ₯2 every Kβ+1β-free graph G on n vertices
with minimum degree Ξ΄(G)>3ββ13ββ4βn is β-partite.
We give a hypergraph version of it. The \emph{minimum positive co-degree} of an
r-graph H, denoted by Ξ΄rβ1+β(H), is the
maximum k such that if S is an (rβ1)-set contained in a edge of
H, then S is contained in at least k distinct edges of
H. Let ββ₯3 be an integer and H be a
Kβ+13β-saturated 3-graph on n vertices. We prove that if
either ββ₯4 and Ξ΄2+β(H)>3ββ13ββ7βn; or β=3 and Ξ΄2+β(H)>2n/7, then H is β-partite; and the bound is best possible.
This is the first stability result on minimum positive co-degree for
hypergraphs