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The planar Tu\'{a}n number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is defined as the maximum number of edges in an $n$-vertex $H$-free planar graph. The exact value of $ex_{\mathcal{P}}(n,H)$ remains a mystery when $H$ is large (for…

Let $F$ be a graph. A hypergraph is called Berge-$F$ if it can be obtained by replacing each edge of $F$ by a hyperedge containing it. Let $\mathcal{F}$ be a family of graphs. The Tur\'an number of Berge-$\mathcal{F}$ is the maximum…

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathscr{F}$ as a subgraph. The Tur\'an number, denoted by $ex(n, \mathscr{F})$, is the maximum number of edges in an $n$-vertex…

A classical Tur\'an problem asks for the maximum possible number of edges in a graph of a given order that does not contain a particular graph $H$ as a subgraph. It is well-known that the chromatic number of $H$ is the graph parameter which…

Given a fixed graph H, we say that a graph G is H-free if G does not contain H as a subgraph. The Tur\'an number ex(n, H) of H is the maximum number of edges in an n-vertex H-free graph. The study of Tur\'an number of graphs is a central…

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…

We start a systematic investigation concerning bipartite Tur\'an number for trees. For a graph $F$ and integers $1 \leq a \leq b$ we define: $(i)$\quad $ex_b(a, b, F)$ is the largest number of edges that an $F$-free bipartite graph can have…

The \emph{Tur\'an function} $\ex(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs where the key…

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…

The Tur\'an number of a $k$-uniform hypergraph $H$, denoted by $e{x_k}\left({n;H} \right)$, is the maximum number of edges in any $k$-uniform hypergraph $F$ on $n$ vertices which does not contain $H$ as a subgraph. Let…

Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp.…

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathcal{F}$ as a subgraph. The general Tur\'an number, denoted by $ex(n, H,\mathscr{F})$, is the maximum number of copies of $H$…

The planar Tur\'an number of a graph $H$, denoted $ex_{_\mathcal{P}}(n,H)$, is the maximum number of edges in a planar graph on $n$ vertices without containing $H$ as a subgraph. This notion was introduced by Dowden in 2016 and has…

Given a graph $F$, we define $\operatorname{ex}(G_{n,p},F)$ to be the maximum number of edges in an $F$-free subgraph of the random graph $G_{n,p}$. Very little is known about $\operatorname{ex}(G_{n,p},F)$ when $F$ is bipartite, with…

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a {\emph rainbow copy} of $F$, that is, a copy of $F$ all of whose edges receive a…

Let $\mathcal{F}$ be a family of graphs. The Tur\'{a}n number $ex(n;\mathcal{F})$ is defined to be the maximum number of edges in a graph of order $n$ that is $\mathcal{F}$-free. In 1959, Erd\H{o}s and Gallai determined the Tur\'an number…

Let $\mathcal{H}$ be an $r$-uniform hypergraph and $F$ be a graph. We say $\mathcal{H}$ contains $F$ as a trace if there exists some set $S\subseteq V(\mathcal{H})$ such that $\mathcal{H}|_{S}:=\{E\cap S: E\in E(\mathcal{H})\}$ contains a…

Given a graph $F$, the planar Tur\'an number of $F$, denoted $\text{ex}_{\mathcal{P}}(n, F)$, is the maximum number of edges in an $n$-vertex $F$-free planar graph. Such an extremal graph problem was initiated by Dowden while determining…

Erd\H{o}s proved an upper bound on the number of edges in an $n$-vertex non-Hamiltonian graph with given minimum degree and showed sharpness via two members of a particular graph family. F\"{u}redi, Kostochka and Luo showed that these two…

Given graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs. The friendship graph $F_k$ consists of $k$ triangles sharing a common vertex. In this…