Related papers: Some sharp results on the generalized Tur\'an numb…

The Tur\'{a}n number of a graph $H$, denoted $\mbox{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph with no subgraph isomorphic to $H$. Solymosi conjectured that if $H$ is any graph and $\mbox{ex}(n,H) = O(n^{\alpha})$…

The spectral Tur\'an theorem states that the $k$-partite Tur\'an graph is the unique graph attaining the maximum adjacency spectral radius among all graphs of order $n$ containing no the complete graph $K_{k+1}$ as a subgraph. This result…

The classical extremal function for a graph $H$, $ex(K_n, H)$ is the largest number of edges in a subgraph of $K_n$ that contains no subgraph isomorphic to $H$. Note that defining $ex(K_n, H-ind)$ by forbidding induced subgraphs isomorphic…

Let $\mathcal{F}$ be a family of graphs. A graph is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$. Generalized Tur\'{a}n problems aim to maximize the number of copies of a graph $H$ in an $n$-vertex…

An extremal graph for a given graph $H$ is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $s,t$ be integers and let $H_{s,t}$ be a graph consisting of $s$ triangles and $t$ cycles of odd…

Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability Theorem of Erd\H{o}s and Simonovits shows that if a $K_{r+1}$-free graph with $n$ vertices has close to the maximal $t_r(n)$ edges, then it is close…

A long-standing conjecture of Erd\H{o}s and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph $H$ such that $\ex(n,H)=\Theta(n^r)$. So far this conjecture is known to be true only for rationals of…

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. The random Tur\'an number $\mathrm{ex}(G^r_{n,p},\mathcal{F})$ is the maximum number of edges in an $\mathcal{F}$-free subgraph of $G^r_{n,p}$, where $G^r_{n,p}$ is the…

The Tur\'an problem asks for the largest number of edges in an $n$-vertex graph not containing a fixed forbidden subgraph $F$. We construct a new family of graphs not containing $K_{s,t}$, for $t= C^s$, with $\Omega(n^{2-1/s})$ edges…

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…

For a $k$-uniform hypergraph $\mathcal{H}$, the \emph{codegree squared sum} $\text{co}_2(\mathcal{H})$ is the square of the $\ell_2$-norm of the codegree vector of $\mathcal{H}$, and for a family $\mathscr{F}$ of $k$-uniform hypergraphs,…

The planar Tur\'{a}n number of a given graph $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges over all planar graphs on $n$ vertices that do not contain a copy of $H$ as a subgraph. Let $H_k$ be a friendship graph,…

For positive integers $s,t,r$, let $K_{s,t}^{(r)}$ denote the $r$-uniform hypergraph whose vertex set is the union of pairwise disjoint sets $X,Y_1,\dots,Y_t$, where $|X| = s$ and $|Y_1| = \dots = |Y_t| = r-1$, and whose edge set is…

For a graph $H$ and a $k$-chromatic graph $F,$ if the Tur\'an graph $T_{k-1}(n)$ has the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough), then $H$ is called $F$-Tur\'an-good, or $k$-Tur\'an-good…

Given a positive integer $n$ and an $r$-uniform hypergraph (or $r$-graph for short) $F$, the Turan number $ex(n,F)$ of $F$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain $F$ as a subgraph. The extension…

We study the generalized Ramsey--Tur\'an function $\mathrm{RT}(n,K_s,K_t,o(n))$, which is the maximum possible number of copies of $K_s$ in an $n$-vertex $K_t$-free graph with independence number $o(n)$. The case when $s=2$ was settled by…

The topological Tur\'an number $\mathrm{ex}_{\hom}(n,X)$ of a 2-dimensional simplicial complex $X$ asks for the maximum number of edges in an $n$-vertex 3-uniform hypergraph containing no triangulation of $X$ as a subgraph. We prove that…

For a simple graph $F$, let $\mathrm{Ex}(n, F)$ and $\mathrm{Ex_{sp}}(n,F)$ denote the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the…

In this paper, we investigate the Tur\'an exponent for $1$-subdivisions of graphs that are neither bipartite nor complete. Specifically, we establish an upper bound on the Tur\'an number of the 1-subdivision of $K_{s,t}^+$, where…

Given a graph $H$ and a positive integer $n,$ the Tur\'{a}n number of $H$ for the order $n,$ denoted ${\rm ex}(n,H),$ is the maximum size of a simple graph of order $n$ not containing $H$ as a subgraph. The book with $p$ pages, denoted…