Related papers: Generalized Tur\'an densities in the hypercube

The Tur\'{a}n number of a graph $H$, $ex(n,H)$, is the maximum number of edges in any graph of order $n$ which does not contain $H$ as a subgraph. Lidick\'{y}, Liu and Palmer determined $ex(n, F_m)$ for $n$ sufficiently large and proved…

Given two $r$-uniform hypergraphs $G$ and $H$ the Tur\'an number $\rm{ex}(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm{ex}(G, H)$ when $G=G_{n,p}^{(r)}$, the Erd\H{o}s-R\'enyi…

Fix a graph $F$. We say that a graph is {\it $F$-free} if it does not contain $F$ as a subgraph. The {\it Tur\'an number} of $F$, denoted $\mathrm{ex}(n,F)$, is the maximum number of edges possible in an $n$-vertex $F$-free graph. The study…

For a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal{F}$ as a subgraph. The maximum number of edges in an $n$-vertex connected…

For a family of graphs $\cal F$, a graph $G$ is $\cal F$-free if it does not contain a member of $\cal F$ as a subgraph. The Tur\'an number $\textrm{ex}(n,{\cal F})$ is the maximum number of edges in an $n$-vertex graph which is $\cal…

Given a graph $H$ and a positive integer $n$, the {\it Tur\'an number} $\ex(n,H)$ is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. A real number $r\in(1,2)$ is called a {\it Tur\'an exponent} if…

Let $\mathcal{H}$ be a family of graphs. The generalized Tur\'an number $ex(n, K_r, \mathcal{H})$ is the maximum number of copies of the clique $K_r$ in any $n$-vertex $\mathcal{H}$-free graph. In this paper, we determine the value of…

The Tur\'an number of a graph $H$, denoted by $ex(n, H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. Let $P_{\ell}$ denote the path on $\ell$ vertices, $S_{\ell-1}$ denote the star on $\ell$…

An $r$-graph is an $r$-uniform hypergraph tree (or $r$-tree) if its edges can be ordered as $E_1,\ldots, E_m$ such that $\forall i>1 \, \exists \alpha(i)<i$ such that $E_i\cap (\bigcup_{j=1}^{i-1} E_j)\subseteq E_{\alpha(i)}$. The Tur\'an…

Given a graph $F$, the $r$-expansion $F^r$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by inserting $r-2$ new distinct vertices in each edge of $F$. Given $r$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{F}$, the generalized…

Given a family $\mathcal{F}$ of $r$-graphs, the Tur\'{a}n number of $\mathcal{F}$ for a given positive integer $N$, denoted by $ex(N,\mathcal{F})$, is the maximum number of edges of an $r$-graph on $N$ vertices that does not contain any…

Given a $k$-graph $H$ a complete blow-up of $H$ is a $k$-graph $\hat{H}$ formed by replacing each $v\in V(H)$ by a non-empty vertex class $A_v$ and then inserting all edges between any $k$ vertex classes corresponding to an edge of $H$.…

The bipartite Tur\'{a}n number of a graph $H$, denoted by $ex(m,n; H)$, is the maximum number of edges in any bipartite graph $G=(X,Y; E)$ with $|X|=m$ and $|Y|=n$ which does not contain $H$ as a subgraph. In this paper, we determined…

The Tur\'{a}n number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a vertex $v$ and a multi-set $\mathcal{F}$ of graphs, the suspension $\mathcal{F}+v$…

Given two graphs $H$ and $F$, the generalized planar Tur\'an number $\mathrm{ex}_\mathcal{P}(n,H,F)$ is the maximum number of copies of $H$ that an $n$-vertex $F$-free planar graph can have. We investigate this function when $H$ and $F$ are…

Given graphs $T$ and $H$, the generalized Tur\'an number ex$(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex graph with no copies of $H$. Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of…

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

Let $H$ be a graph on $n$ vertices and let the blow-up graph $G[H]$ be defined as follows. We replace each vertex $v_i$ of $H$ by a cluster $A_i$ and connect some pairs of vertices of $A_i$ and $A_j$ if $(v_i,v_j)$ was an edge of the graph…

In the so-called generalized Tur\'an problems we study the largest number of copies of $H$ in an $n$-vertex $F$-free graph $G$. Here we introduce a variant, where $F$ is not forbidden, but we restrict how copies of $H$ and $F$ can be placed…

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…