Related papers: Tur\'an Density of $2$-edge-colored Bipartite Grap…

For a fixed positive integer $n$ and an $r$-uniform hypergraph $H$, the Tur\'an number $ex(n,H)$ is the maximum number of edges in an $H$-free $r$-uniform hypergraph on $n$ vertices, and the Lagrangian density of $H$ is defined as…

Denote by $\mathcal{C}^-_{\ell}$ the $3$-uniform hypergraph obtained by removing one hyperedge from the tight cycle on $\ell$ vertices. It is conjectured that the Tur\'an density of $\mathcal{C}^-_{5}$ is $1/4$. In this paper, we make…

We say a hypergraph $\mathcal{H}$ contains a graph $G$ as trace if there exists a vertex subset $S \subseteq V(\mathcal{H})$ such that $|S| = V(G)$ and $\{e \cap S \mid e \in E(\mathcal{H})\}$ contains $G$ as a subgraph. We use…

Write $K^{(k)}_{n}$ for the complete $k$-graph on $n$ vertices. For $2 \leq k \leq g < r$ integers, let $\pi\left(n, K^{(k)}_{g}, K^{(k)}_r\right)$ be the maximum density of $K^{(k)}_{g}$ in $n$ vertex $K^{(k)}_{r}$-free $k$-graphs. The…

The Tur\'an number of a graph $H$, denoted by $\text{ex}(n, H)$, is the maximum number of edges in an $n$-vertex graph that does not have $H$ as a subgraph. Let $TP_k$ be the triangular pyramid of $k$-layers. In this paper, we determine…

The $r$-expansion $G^+$ of a graph $G$ is the $r$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a vertex subset of size $r-2$ disjoint from $V(G)$ such that distinct edges are enlarged by disjoint subsets. Let…

We study the generalized Tur\'an function $ex(n,H,F)$, when $H$ or $F$ is a double star $S_{a,b}$, which is a tree with a central edge $uv$, $a$ leaves connected to $u$ and $b$ leaves connected to $v$. We determine $ex(n,K_k,S_{a,b})$ and…

The Tur\'an number of an r-uniform hypergraph H is the maximum number of edges in any r-graph on n vertices which does not contain H as a subgraph. Let P_l^(r) denote the family of r-uniform loose paths on l edges, F(k,l) denote the family…

The {\em Tur\'an number} of an $r$-uniform graph $F$, denoted by $ex(n,F)$, is the maximum number of edges in an $F$-free $r$-uniform graph on $n$ vertices. The {\em Tur\'{a}n density} of $F$ is defined as…

For two graphs $F$ and $H$, the relative Tur\'{a}n number $\mathrm{ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau \cite{FKP} and Perarnau and Reed \cite{PR} studied these…

The Tur\'an number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. A wheel $W_n$ is an $n$-vertex graph formed by connecting a single vertex to all vertices…

For $k\ge 3$, the $(k-2)$-uniform Tur\'an density $\pi_{k-2}(F)$ of a $k$-graph $F$ is the supremum of $d$ for which there are arbitrarily large $F$-free $k$-graphs that are uniformly $d$-dense with respect to the $k$-vertex cliques of…

For two $s$-uniform hypergraphs $H$ and $F$, the Tur\'{a}n number $ex_s(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Let $s, r, k, n_1, \ldots, n_r$ be integers satisfying $2\leq s\leq r$ and $n_1\leq n_2\leq…

Given $p\geq 0$ and a graph $G$ whose degree sequence is $d_1,d_2,\ldots,d_n$, let $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\'an-type problem for $e_p(G)$: given $p\geq 0$, how large can $e_p(G)$ be if $G$ has no…

The Tur\'{a}n number of a graph $H$, $ex(n,H)$, is the maximum number of edges in a simple graph of order $n$ which does not contain $H$ as a subgraph. Let $k\cdot P_3$ denote $k$ disjoint copies of a path on $3$ vertices. In this paper, we…

The Turan number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. We determine the Turan number and find the unique extremal graph for forests consisting of paths when n is…

Let $k\geq 3$. Given a $k$-uniform hypergraph $H$, the minimum codegree $\delta(H)$ is the largest $d\in\mathbb{N}$ such that every $(k-1)$-set of $V(H)$ is contained in at least $d$ edges. Given a $k$-uniform hypergraph $F$, the codegree…

For graphs $H_1$ and $H_2$, if we glue them by identifying a given pair of vertices $u \in V(H_1)$ and $v \in V(H_2)$, what is the extremal number of the resulting graph $H_1^u \odot H_2^v$? In this paper, we study this problem and show…

The Tur\'an hypergraph problem asks to find the maximum number of $r$-edges in a $r$-uniform hypergraph on $n$ vertices that does not contain a clique of size $a$. When $r=2$, i.e., for graphs, the answer is well-known and can be found in…

Let $\mathrm{ex}(n, H)$ be the Tur\'{a}n number of $H$ for a given graph $H$. A graph is color-critical if it contains an edge whose removal reduces its chromatic number. Simonovits' chromatic critical edge theorem states that if $H$ is…