Related papers: $\ell$-degree Tur\'an density

Let $\mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $\mathrm{rex}(n, F)$, that are best possible up to a constant factor, when…

For $k \ge 1$ an integer, a set $S$ of vertices in a graph $G$ with minimum degree at least~$k-1$ is a $k$-tuple dominating set of $G$ if every vertex of $S$ is adjacent to at least $k-1$ vertices in $S$ and every vertex of $V(G) \setminus…

Let $\mathcal{Q}_n$ be the $n$-dimensional hypercube: the graph with vertex set $\{0,1\}^n$ and edges between vertices that differ in exactly one coordinate. For $1\leq d\leq n$ and $F\subseteq \{0,1\}^d$ we say that $S\subseteq \{0,1\}^n$…

The Tur\'{a}n number $ex(n,H)$ of a graph $H$ is the maximum number of edges in any $H$-free graph on $n$ vertices. The triangular pyramid of $k$-layers, denoted by $TP_k$, is a generalization of a triangle. The Tur\'an problems of a…

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\'{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 $\mbox{ex}(n,H) = O(n^{\alpha})$…

Let $F$ be a strictly balanced $r$-uniform hypergraph with $e>2$ edges and $r$-density $m$. We give a new short proof of the fact that the Tur\'an number $\ex(n, F)$ is greater than $c\, n^{r-1/m} (\log n)^{1/(e-1)}$ where $c$ depends only…

Given an $r$-graph $F$ with $r \ge 2$, let $\mathrm{ex}(n, (t+1) F)$ denote the maximum number of edges in an $n$-vertex $r$-graph with at most $t$ pairwise vertex-disjoint copies of $F$. Extending several old results and complementing…

An $r$-pattern $P$ is defined as an ordered pair $P=([l],E)$, where $l$ is a positive integer and $E$ is a set of $r$-multisets with elements from $[l]$. An $r$-graph $H$ is said to be $P$-colorable if there is a homomorphism $\phi$:…

A tight $\ell$-cycle minus an edge $C_\ell^-$ is the $3$-graph on the vertex set $[\ell]$, where any three consecutive vertices in the string $123\ldots\ell 1$ form an edge. We show that for every $\ell\ge 5$, $\ell$ not divisible by $3$,…

The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem states that any $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ has at most $O_s(n^{2-1/s})$ edges. In the past two decades, motivated by the applications in…

Let the tight $\ell$-cycle minus one edge $C_\ell^{3-}$ be the $3$-graph on $\{1,\dots,\ell\}$ consisting of $\ell-1$ consecutive triples in the cyclic order. We show that, for every $\ell\ge 5$ not divisible by $3$, the Tur\'an density 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 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…

Let $G$ be an $r$-uniform hypergraph on $n$ vertices such that all but at most $\varepsilon \binom{n}{\ell}$ $\ell$-subsets of vertices have degree at least $p \binom{n-\ell}{r-\ell}$. We show that $G$ contains a large subgraph with high…

Let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-set $[n]=\{1,2,\ldots,n\}$. Let $n>2k$ and let $\mathcal{F}\subset \binom{[n]}{k}$ be an {\it intersecting} $k$-graph, i.e., $F\cap F'\neq \emptyset$ for all…

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…

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…

For fixed graphs $H$ and $F$, the \emph{generalized Tur\'an number} $\mathrm{ex}(n,H,F)$ is the maximum possible number of copies of a subgraph $H$ in an $n$-vertex $F$-free graph. This article is a survey of this extremal function whose…

For positive integers $n\ge s> r$, the Tur\'an function $T(n,s,r)$ is the smallest size of an r-graph with n vertices such that every set of s vertices contains at least one edge. Also, define the Tur\'an density $t(s,r)$ as the limit of…