相关论文: On a theorem of Erd\H{o}s and Simonovits on graphs…

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…

In 1964, Erd\H{o}s proposed the problem of estimating the Tur\'an number of the $d$-dimensional hypercube $Q_d$. Since $Q_d$ is a bipartite graph with maximum degree $d$, it follows from results of F\"uredi and Alon, Krivelevich, Sudakov…

The classical stability theorem of Erd\H{o}s and Simonovits states that, for any fixed graph with chromatic number $k+1 \ge 3$, the following holds: every $n$-vertex graph that is $H$-free and has within $o(n^2)$ of the maximal possible…

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…

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 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 equal to a positive proportion of the…

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…

Given a graph $L$, the Tur\'an number $\textrm{ex}(n,L)$ is the maximum possible number of edges in an $n$-vertex $L$-free graph. The study of Tur\'an number of graphs is a central topic in extremal graph theory. Although the celebrated…

Let the crown $C_{13}$ be the linear $3$-graph on $9$ vertices $\{a,b,c,d,e,f,g,h,i\}$ with edges $$E = \{\{a,b,c\}, \{a, d,e\}, \{b, f, g\}, \{c, h,i\}\}.$$ Proving a conjecture of Gy\'arf\'as et. al., we show that for any crown-free…

The following sharpening of Tur\'an's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there…

For ordinary graphs it is known that any graph $G$ with more edges than the Tur{\'a}n number of $K_s$ must contain several copies of $K_s$, and a copy of $K_{s+1}^-$, the complete graph on $s+1$ vertices with one missing edge. Erd\H{o}s…

Let $k \ge 2$ be an integer. We show that if $s = 2$ and $t \ge 2$, or $s = t = 3$, then the maximum possible number of edges in a $C_{2k+1}$-free graph containing no induced copy of $K_{s,t}$ is asymptotically equal to $(t - s +…

Let $\mc{F}$ be a family of graphs. A graph is {\em $\mc{F}$-free} if it contains no copy of a graph in $\mc{F}$ as a subgraph. A cornerstone of extremal graph theory is the study of the {\em Tur\'an number} $ex(n,\mc{F})$, the maximum…

The Tur\'an 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. Let P_k be the path with k vertices, the square P^2_k of P_k is obtained by joining the pairs of vertices…

In 1975, Erd\H{o}s asked for the maximum number of edges that an $n$-vertex graph can have if it does not contain two edge-disjoint cycles on the same vertex set. It is known that Tur\'an-type results can be used to prove an upper bound of…

We consider finite simple graphs. 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 graph of order $n$ not containing $H$ as a subgraph. Erd\H{o}s…

Let $F_{3,3}$ be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. We show that for all $n \ge 6$, the maximum…

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…

The generalized Tur\'{a}n number ${\rm ex}(G,H)$ is the maximum number of edges in an $H$-free subgraph of a graph $G.$ It is an important extension of the classical Tur\'{a}n number ${\rm ex}(n,H)$, which is the maximum number of edges in…

A conjecture of Erd\H{o}s from 1967 asserts that any graph on $n$ vertices which does not contain a fixed $r$-degenerate bipartite graph $F$ has at most $Cn^{2-1/r}$ edges, where $C$ is a constant depending only on $F$. We show that this…