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We investigate the impact of a high-degree vertex in Tur\'{a}n problems for degenerate hypergraphs (including graphs). We say an $r$-graph $F$ is bounded if there exist constants $\alpha, \beta>0$ such that for large $n$, every $n$-vertex $F$-free $r$-graph with a vertex of degree at least $\alpha \binom{n-1}{r-1}$ has fewer than $(1-\beta) \cdot \mathrm{ex}(n,F)$ edges. The boundedness property is crucial for recent works~\cite{HHLLYZ23a,DHLY24} that aim to extend the classical Hajnal--Szemer\'{e}di Theorem and the anti-Ramsey theorems of Erd\H{o}s--Simonovits--S\'{o}s. We show that many well-studied degenerate hypergraphs, such as all even cycles, most complete bipartite graphs, and the expansion of most complete bipartite graphs, are bounded. In addition, to prove the boundedness of the expansion of complete bipartite graphs, we introduce and solve a Zarankiewicz-type problem for $3$-graphs, strengthening a theorem by Kostochka--Mubayi--Verstra\"{e}te~\cite{KMV15}., Comment: comments are welcome

Unlike graphs, determining Tur\'{a}n densities of hypergraphs is known to be notoriously hard in general. The essential reason is that for many classical families of $r$-uniform hypergraphs $\mathcal{F}$, there are perhaps many near-extremal $\mathcal{M}_t$-free configurations with very different structure. Such a phenomenon is called not stable, and Liu and Mubayi gave a first not stable example. Another perhaps reason is that little is known about the set consisting of all possible Tur\'{a}n densities which has cardinality of the continuum. Let $t\ge 2$ be an integer. In this paper, we construct a finite family $\mathcal{M}_t$ of 3-uniform hypergraphs such that the Tur\'{a}n density of $\mathcal{M}_t$ is irrational, and there are $t$ near-extremal $\mathcal{M}_t$-free configurations that are far from each other in edit-distance. This is the first not stable example that has an irrational Tur\'{a}n density. It also provides a new phenomenon about feasible region functions.

For a nondegenerate $r$-graph $F$, large $n$, and $t$ in the regime $[0, c_{F} n]$, where $c_F>0$ is a constant depending only on $F$, we present a general approach for determining the maximum number of edges in an $n$-vertex $r$-graph that does not contain $t+1$ vertex-disjoint copies of $F$. In fact, our method results in a rainbow version of the above result and includes a characterization of the extremal constructions. Our approach applies to many well-studied hypergraphs (including graphs) such as the edge-critical graphs, the Fano plane, the generalized triangles, hypergraph expansions, the expanded triangles, and hypergraph books. Our results extend old results of Simonovits~\cite{SI68} and Moon~\cite{Moon68} on complete graphs and can be viewed as a step towards a general density version of the classical Corr\'{a}di--Hajnal Theorem~\cite{CH63}., Comment: corrected some typo and added a theorem (THM 2.7) about expanded Erdos-Sos trees

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.

Let $r\ge 2$ be an integer. The real number $\alpha\in [0,1]$ is a jump for $r$ if there exists a constant $c > 0$ such that for any $\epsilon >0$ and any integer $m \geq r$, there exists an integer $n_0(\epsilon, m)$ satisfying any $r$-uniform graph with $n\ge n_0(\epsilon, m)$ vertices and density at least $\alpha+\epsilon$ contains a subgraph with $m$ vertices and density at least $\alpha+c$. A result of Erd\H{o}s, Stone and Simonovits implies that every $\alpha\in [0,1)$ is a jump for $r=2$. Erd\H{o}s asked whether the same is true for $r\ge 3$. Frankl and R\"{o}dl gave a negative answer by showing that $1-\frac{1}{l^{r-1}}$ is not a jump for $r$ if $r\ge 3$ and $l>2r$. After that, more non-jumps are found using a method of Frankl and R\"{o}dl. In this note, we show a method to construct maps $f \colon [0,1] \to [0,1]$ that preserve non-jumps, if $\alpha$ is a non-jump for $r$ given by the method of Frankl and R\"{o}dl, then $f(\alpha)$ is also a non-jump for $r$. We use these maps to study hypergraph Tur\'{a}n densities and answer a question posed by Grosu.

We give the first exact and stability results for a hypergraph Tur\'{a}n problem with infinitely many extremal constructions that are far from each other in edit-distance. This includes an example of triple systems with Tur\'{a}n density $2/9$, thus answering some questions posed by the third and fourth authors and Reiher about the feasible region of hypergraphs. Our results also provide extremal constructions whose shadow density is a transcendental number. Our novel approach is to construct certain multilinear polynomials that attain their maximum (in the standard simplex) on a line segment and then to use these polynomials to define an operation on hypergraphs that gives extremal constructions., Comment: journal version, Discrete Analysis 2023:18, 34 pp