Two Stability Theorems for K ℓ + 1 r -Saturated Hypergraphs

Let $ℱ$ be a family of $r$-uniform hypergraphs (henceforth $r$-graphs). An $ℱ$-saturated $r$-graph is a maximal $r$-graph not containing any member of $ℱ$ as a subgraph. For integers $\ell \ge r\ge 2$, let $K_{\ell +1}^r$ be the collection of all $r$-graphs $F$ with at most $\left(\frac{\ell +1}{2}\right)$ edges such that for some $\left(\ell +1\right)$-set $S$ every pair $\left\{u,v\right\}\subset S$ is covered by an edge in $F$, and let $T_r\left(n,\ell \right)$ be the complete $\ell$-partite $r$-graph on $n$ vertices with no two part sizes differing by more than one. Let $t_r\left(n,\ell \right)$ be the number of edges in $T_r\left(n,\ell \right)$. Our first result shows that for each $\ell \ge r\ge 2$ every $K_{\ell +1}^r$-saturated $r$-graph on $n$ vertices with $t_r\left(n,\ell \right)-o\left(n^{r-1+1∕\ell }\right)$ edges contains a complete $\ell$-partite subgraph on $\left(1-o\left(1\right)\right)n$ 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 Andrásfai, Erdős, and Sós, which states that for $\ell \ge 2$ every $K_{\ell +1}$-free graph $G$ on $n$ vertices with minimum degree $\delta \left(G\right)>\frac{3\ell -4}{3\ell -1}n$ is $\ell$-partite. We give a hypergraph version of it. The minimum positive co-degree of an $r$-graph $\mathscr{H}$, denoted by ${\delta }_{r-1}^{+}\left(\mathscr{H}\right)$, is the maximum $k$ such that if $S$ is an $\left(r-1\right)$-set contained in an edge of $\mathscr{H}$, then $S$ is contained in at least $k$ distinct edges of $\mathscr{H}$. Let $\ell \ge 3$ be an integer and $\mathscr{H}$ be a $K_{\ell +1}^3$-saturated 3-graph on $n$ vertices. We prove that if either $\ell \ge 4$ and ${\delta }_2^{+}\left(\mathscr{H}\right)>\frac{3\ell -7}{3\ell -1}n$; or $\ell =3$ and ${\delta }_2^{+}\left(\mathscr{H}\right)>2n∕7$, then $\mathscr{H}$ is $\ell$-partite; and the bound is best possible. This is the first stability result on minimum positive co-degree for hypergraphs.