Related papers: On local Tur\'an problems

In the early 1980s, Erd\H{o}s and S\'os initiated the study of the classical Tur\'an problem with a uniformity condition: the uniform Tur\'an density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large…

In the 1980s, Erd\H{o}s and S\'os initiated the study of Tur\'an hypergraph problems with a uniformity condition on the distribution of edges, i.e., determining density thresholds for the existence of a hypergraph H in a host hypergraph…

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…

We describe a method that we believe may be foundational for a comprehensive theory of generalised Turan problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard…

Let $G$ be a cancellative $3$-uniform hypergraph in which the symmetric difference of any two edges is not contained in a third one. Equivalently, a $3$-uniform hypergraph $G$ is cancellative if and only if $G$ is $\{F_4, F_5\}$-free, where…

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…

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…

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$…

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…

We consider the Tur\'an problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ with $E_r$ and $E_b$ do not…

Given graphs $H$ and $F$ with $\chi(H)<\chi(F)$, we say that $H$ is weakly $F$-Tur\'an-good if among $n$-vertex $F$-free graphs, a $(\chi(F)-1)$-partite graph contains the most copies of $H$. Let $H$ be a bipartite graph that contains a…

The generalized Tur\'{a}n number $\mathrm{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. For an integer $t \geq 1$, let $tF$ be the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku,…

Determining the Tur\'an densities of hypergraphs is a notoriously difficult problem at the core of combinatorics. Although Tur\'an posed this problem in 1941, $\pi(K_{\ell}^{(k)})$ remains unknown for all $\ell>k\geq 3$. Prior to this work,…

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich…

Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized Tur\'an number $\mathrm{ex}(n,T,\mathcal{F})$ is the maximum number of copies of $T$ in an $n$-vertex $\mathcal{F}$-free graph. We prove a general theorem which states…

For a forbidden graph $L$, let $ex(p;L)$ denote the maximal number of edges in a simple graph of order $p$ not containing $L$. Let $T_n$ denote the unique tree on $n$ vertices with maximal degree $n-2$, and let $T_n^*=(V,E)$ be the tree on…

Given a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a {\em Berge-$F$} if there is a bijection $\phi:E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-uniform…

Given a family of $k$-hypergraphs $\mathcal{F}$, $ex(n,\mathcal{F})$ is the maximum number of edges a $k$-hypergraph can have, knowing that said hypergraph has $n$ vertices but contains no copy of any hypergraph from $\mathcal{F}$ as a…

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. The $r$-expansion $F^{r}$ of a graph $F$ is the $r$-uniform hypergraph obtained from $F$ by enlarging each edge of $F$ with a vertex subset of size…

Let $\mathcal{H}$ be a hypergraph and $F$ be a graph. If there exists a bijection between the hyperedges of $\mathcal{H}$ and the edges of $F$ such that each hyperedge contains its image, then we say that $\mathcal{H}$ is a \textit{Berge…