Related papers: Every graph is eventually Tur\'an-good

The generalized Tur\'an number for $H$ of $G$, denoted by $\ex(n,H,G)$, is the maximum number of copies of $H$ in an $n$-vertex $G$-free graph. When $H$ is an edge, $\ex(n,H,G)$ is the classical Tur\'an number $\ex(n,G)$. Let $P_k$ be the…

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…

We combine two generalizations of ordinary Tur\'an problems. Given graphs $H$ and $F$ and a positive integer $n$, we study $rex(n, H, F )$, which is the largest number of copies of $H$ in $F$-free regular $n$-vertex graphs.

We provide a deterministic polynomial-time algorithm that, for a given $k$-uniform hypergraph $H$ with $n$ vertices and edge density $d$, finds a complete $k$-partite subgraph of $H$ with parts of size at least ${c(d, k)(\log…

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…

Let $K^{(r)}_{s_1,s_2,\cdots,s_r}$ be the complete $r$-partite $r$-uniform hypergraph and $ex(n,K^{(r)}_{s_1,s_2,\cdots,s_r})$ be the maximum number of edges in any $n$-vertex $K^{(r)}_{s_1,s_2,\cdots,s_r}$-free $r$-uniform hypergraph. It…

The Tur\'an number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. We determine…

As a variant of the famous Tur\'an problem, we study $\mathrm{rex}(n,F)$, the maximum number of edges that an $n$-vertex regular graph can have without containing a copy of $F$. We determine $\mathrm{rex}(n,K_{r+1})$ for all pairs of…

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…

The Tur\'an number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. A wheel $W_n$ is an $n$-vertex graph formed by connecting a single vertex to all vertices…

Given a graph $H$, the Tur\'{a}n number ${\rm ex}(n,H)$ of $H$ is the maximum number of edges of an $n$-vertex simple graph containing no $H$ as a subgraph. Let $kK_p$ denote the disjoint union of $k$ copies of the complete graph $K_p$. In…

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge…

Given two graphs $T$ and $F$, the maximum number of copies of $T$ in an $F$-free graph on $n$ vertices is called the generalized Tur\'{a}n number, denoted by $ex(n,T,F)$. When $T=K_2$, it reduces to the classical Tur\'{a}n number $ex(n,F)$.…

Let $F$ be an $(r+1)$-color critical graph with $r\geq 2$, that is, $\chi(F)=r+1$ and there is an edge $e$ in $F$ such that $\chi(F-e)=r$. Gerbner recently conjectured that every $n$-vertex maximal $F$-free graph with at least…

A classical conjecture of Erd\H{o}s and S\'os asks to determine the Tur\'an number of a tree. We consider variants of this problem in the settings of hypergraphs and multi-hypergraphs. In particular, for all $k$ and $r$, with $r \ge k…

Generalized Tur\'an problems ask for the maximum number of copies of a graph $H$ in an $n$-vertex, $F$-free graph, denoted by ex$(n,H,F)$. We show how to extend the new, localized approach of Brada\v{c}, Malec, and Tompkins to generalized…

Given integers $r \geq 2$, $k \geq 3$ and $2 \leq s \leq \binom{k}{2}$, and a graph $G$, we consider $r$-edge-colorings of $G$ with no copy of a complete graph $K_k$ on $k$ vertices where $s$ or more colors appear, which are called…

We show that if G is a graph of sufficiently large order n containing as many r-cliques as the r-partite Turan graph of order n; then for some C>0 G has more than Cn^(r-1) (r+1)-cliques sharing a common edge unless G is isomorphic to the…

We consider an infinite version of the bipartite Tur\'{a}n problem. Let $G$ be an infinite graph with $V(G) = \mathbb{N}$ and let $G_n$ be the $n$-vertex subgraph of $G$ induced by the vertices $\{1,2, \dots, n \}$. We show that if $G$ is…

The $r$-expansion of a $k$-uniform hypergraph $H$, denoted by $H^{(+r)}$, is an $r$-uniform hypergraph obtained by enlarging each $k$-edge of $H$ with a set of $r-k$ vertices of degree one. The random Tur\'an number…