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For graphs $H$ and $F$, let $\operatorname{ex}(n, H, F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. The study of this function, which generalises the well-studied Tur\'an numbers of graphs, was…

For given graphs $G$ and $F$, the Tur\'an number $ex(G,F)$ is defined to be the maximum number of edges in an $F$-free subgraph of $G$. Foucaud, Krivelevich and Perarnau and later independently Briggs and Cox introduced a dual version of…

Very recently, Alon and Frankl, and Gerbner studied the maximum number of edges in $n$-vertex $F$-free graphs with bounded matching number, respectively. We consider the analogous Tur\'{a}n problems on hypergraphs with bounded matching…

An edge-colored graph is said to contain a rainbow-$F$ if it contains $F$ as a subgraph and every edge of $F$ is a distinct color. The problem of maximizing edges among $n$-vertex properly edge-colored graphs not containing a rainbow-$F$,…

A graph $G$ is rainbow-$F$-free if it admits a proper edge-coloring without a rainbow copy of $F$. The rainbow Tur\'an number of $F$, denoted $\mathrm{ex^*}(n,F)$, is the maximum number of edges in a rainbow-$F$-free graph on $n$ vertices.…

The Tur\'{a}n problem asks for the largest number of edges ex$(n,H)$ in an $n$-vertex graph not containing a fixed forbidden subgraph $H$, which is one of the most important problems in extremal graph theory. However the order of magnitude…

The {\em planar Tur\'{a}n number} of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $H$-free planar graph. The planar Tur\'{a}n number of $k\geq 3$ vertex-disjoint union of cycles is a trivial value $3n-6$.…

The generalized Tur\'an number $\mathrm{ex}(n, H, \mathcal{F})$ is defined as the maximum number of copies of a graph $H$ in an $n$-vertex graph that does not contain any graph $F \in \mathcal{F}$. Alon and Frankl initiated the study of…

An edge colored graph is said to contain rainbow-$F$ if $F$ is a subgraph and every edge receives a different color. In 2007, Keevash, Mubayi, Sudakov, and Verstra\"ete introduced the \emph{rainbow extremal number} $\mathrm{ex}^*(n,F)$, a…

In this paper we show that for $r\geq 12$ and any sufficiently large $n$-vertex graph $G$ the number of $r$-edge-colorings of $G$ with no rainbow $K_4$ is at most $r^{ex(n,K_4)}$, where $ex(n,K_4)$ denotes the Tur\'{a}n number of $K_4$.…

Given a hypergraph $\mathcal{H}$ and a graph $G$, we say that $\mathcal{H}$ is a \textit{Berge}-$G$ if there is a bijection between the hyperedges of $\mathcal{H}$ and the edges of $G$ such that each hyperedge contains its image. We denote…

For graphs $T, H$, let $ex(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex $H$-free graph. In this paper we prove some sharp results on this generalization of Tur\'an numbers, where our focus is for the graphs $T,H$…

Tur\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geq 2$ every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually…

It is well known that spectral Tur\'{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur\'{a}n type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we…

The extremal number $\mathrm{ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with $\mathrm{ex}(n , F) =…

A {\it vertex-ordered} graph is a graph equipped with a linear ordering of its vertices. A pair of independent edges in an ordered graph can exhibit one of the following three patterns: separated, nested or crossing. We say a pair of…

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. The $r$-expansion $F^{r}$ of a graph $F$ is the $r$-uniform hypergraph obtained from $F$ by enlarging each edge of $F$ with a vertex subset of size…

Let $H_{s,t_1,\ldots ,t_k}$ be the graph with $s$ triangles and $k$ odd cycles of lengths $t_1,\ldots ,t_k\ge 5$ intersecting in exactly one common vertex. Recently, Hou, Qiu and Liu [Discrete Math. 341 (2018) 126--137], and Yuan [J. Graph…

Given a graph $H$ and a positive integer $n$, the Tur\'{a}n number of $H$ for the order $n$, denoted $ex(n,H)$, is the maximum size of a simple graph of order $n$ not containing $H$ as a subgraph. Given graphs $G$ and $H$, the notation $G…

A classical extremal, or Tur\'an-type problem asks to determine ${\rm ex}(G, H)$, the largest number of edges in a subgraph of a graph $G$ which does not contain a subgraph isomorphic to $H$. Alon and Shikhelman introduced the so-called…