Related papers: On Relative Ordered Tur\'an Density

For a given collection $\mathcal{G} = (G_1,\dots, G_k)$ of graphs on a common vertex set $V$, which we call a \emph{graph system}, a graph $H$ on a vertex set $V(H) \subseteq V$ is called a \emph{rainbow subgraph} of $\mathcal{G}$ if there…

A graph is cubical if it is a subgraph of a hypercube. For a cubical graph $H$ and a hypercube $Q_n$, $ex(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $ex(Q_n, H)$ is at least a positive proportion of the…

We study the following generalization of the Tur\'an problem in sparse random graphs. Given graphs $T$ and $H$, let $\mathrm{ex}\big(G(n,p), T, H\big)$ be the random variable that counts the largest number of copies of $T$ in a subgraph of…

Given graphs $H$ and $F$, the generalized Tur\'an number $\ex(n, H, F)$ is defined as the maximum number of copies of $H$ in an $n$-vertex graph that contains no copy of $F$. The suspension $\widehat{F}$ of a graph $F$ is obtained by adding…

Given a set of graphs $\mathcal{H}$, we say that a graph $G$ is \textit{$\mathcal{H}$-free} if it does not contain any member of $\mathcal{H}$ as a subgraph. Let $\text{ex}(n,\mathcal{H})$ (resp. $\text{ex}_{sp}(n,\mathcal{H})$) denote the…

The Tur\'an number $\text{ex}(n,H)$ of a graph $H$ is the maximal number of edges in an $H$-free graph on $n$ vertices. In $1983$ Chung and Erd\H{o}s asked which graphs $H$ with $e$ edges minimize $\text{ex}(n,H)$. They resolved this…

Fix a $k$-chromatic graph $F$. In this paper we consider the question to determine for which graphs $H$ does the Tur\'an graph $T_{k-1}(n)$ have the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large…

The Tur\'{a}n number of a graph $H$ is the maximum number of edges in any graph of order $n$ that does not contain $H$ as a subgraph. In 1959, Erd\H os and Gallai obtained a sharp upper bound of Tur\'{a}n numbers for a path of arbitrary…

The generalized Tur\'{a}n number $ex(n,K_s,H)$ is the maximum number of complete graph $K_s$ in an $H$-free graph on $n$ vertices. Let $F_k$ be the friendship graph consisting of $k$ triangles. Erd\H{o}s and S\'os (1976) determined the…

Given a graph $F$, we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$. We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper…

The $3$-uniform tight $\ell$-cycle $C_\ell^{3}$ is the $3$-graph on $\{1,\dots,\ell\}$ consisting of all $\ell$ consecutive triples in the cyclic order. Let $\mathcal{C}$ be either the pair $\{C_{4}^{3}, C_{5}^{3}\}$ or the single tight…

A real number $\alpha\in [0, 1)$ is a jump for an integer $r\ge 2$ if there exists $c>0$ such that no number in $(\alpha , \alpha + c)$ can be the Tur\'an density of a family of $r$-uniform graphs. A classical result of Erd\H os and Stone…

Given a graph $T$ and a family of graphs $\mathcal{H}$. The generalized Tur\'an number of $\mathcal{H}$ is the maximum number of copies of $T$ in an $\mathcal{H}$-free graph on $n$ vertices, denoted by $ex(n, T, \mathcal{H})$. Let $ex(n, T,…

We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in…

In this paper, we consider the Tur\'an problems on $\{1,3\}$-hypergraphs. We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set $V=[5]$ and…

For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $r\in \mathbb{N}$, the $r$-independence number of $G$, denoted $\alpha_r(G)$ is the largest size of a…

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The Tur\'{a}n number $\text{ex}(n,\mathcal{H})$ is the maximum number of edges in an $n$-vertex $\mathcal{H}$-free $r$-uniform hypergraph. The Tur\'{a}n density of $\mathcal{H}$ is defined by…

The generalized Tur\'an number $\ex(n,K_s,F)$ denotes the maximum number of copies of $K_s$ in an $n$-vertex $F$-free graph. Let $kF$ denote $k$ disjoint copies of $F$. Gerbner, Methuku and Vizer [DM, 2019, 3130-3141] gave a lower bound for…

For graphs $F$ and $H$, let $i(F)$ denote the inducibility of $F$ and let $i_H(F)$ denote the inducibility of $F$ over $H$-free graphs. We prove that for almost all graphs $F$ on a given number of vertices, $i_{K_k}(F)$ attains infinitely…

For two graphs $G$ and $H$, the Tur\'{a}n number $ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Tur\'{a}n numbers $ex(K_{m,n},kK_2)$ for all $k\geq 1$ [7]. In this…