Related papers: 3-uniform hypergraphs with few Berge paths of leng…

Since its formulation, Tur\'an's hypergraph problems have been among the most challenging open problems in extremal combinatorics. One of them is the following: given a $3$-uniform hypergraph $\mathcal{F}$ on $n$ vertices in which any five…

An ordered hypergraph is a hypergraph $G$ whose vertex set $V(G)$ is linearly ordered. We find the Tur\'an numbers for the $r$-uniform $s$-vertex tight path $P^{(r)}_s$ (with vertices in the natural order) exactly when $r\le s < 2r$ and $n$…

Reiher, R\"odl and Schacht [J. London Math. Soc. 97 (2018), 77--97] showed that the uniform Tur\'an density of every $3$-uniform hypergraph is either $0$ or at least $1/27$, and asked whether there exist $3$-uniform hypergraphs with uniform…

Given $r$-uniform hypergraphs $G$ and $H$ the Tur\'an number $\rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the Erd\H{o}s-R\'enyi random…

We expand Conlon's random algebraic construction to show that for any odd number $k \geq 3$ exists a natural number $c_k$ (the same as Conlon's) such that $\operatorname{ex}(n^a,n,\theta_{k,c_k}) = \Omega_{k,a}((n^{1 + a})^{\frac{k +…

An $r$-uniform hypergraph ($r$-graph for short) is linear if any two edges intersect at most one vertex. Let $\mathcal{F}$ be a given family of $r$-graphs. An $r$-graph $H$ is called $\mathcal{F}$-free if $H$ does not contain any member of…

The $3$-uniform tight $\ell$-cycle minus one edge $C_{\ell}^{3-}$ is the $3$-graph on $\ell$ vertices consisting of $\ell-1$ consecutive triples in the cyclic order. We show that for every integer $\ell \ge 5$ satisfying $\ell\not\equiv…

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

We give the first exact and stability results for a hypergraph Tur\'{a}n problem with infinitely many extremal constructions that are far from each other in edit-distance. This includes an example of triple systems with Tur\'{a}n density…

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…

Let $ a,b \in {\bf Z}^+$, $r=a + b$, and let $T$ be a tree with parts $U = \{u_1,u_2,\dots,u_s\}$ and $V = \{v_1,v_2,\dots,v_t\}$. Let $U_1, \dots ,U_s$ and $V_1, \dots, V_t$ be disjoint sets, such that {$|U_i|=a$ and $|V_j|=b$ for all…

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…

The Tur\'an number of a $k$-uniform hypergraph $H$, denoted by $e{x_k}\left({n;H} \right)$, is the maximum number of edges in any $k$-uniform hypergraph $F$ on $n$ vertices which does not contain $H$ as a subgraph. Let…

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…

Let $F$ be a graph. We say that a hypergraph $H$ is a {\it Berge}-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The…

An $r$-uniform hypergraph ($r$-graph) is linear if any two edges intersect at most one vertex. For a graph $F$, a hypergraph $H$ is Berge-$F$ if there is a bijection $\phi:E(F)\rightarrow E(H)$ such that $e\subseteq \phi(e)$ for all $e$ in…

A Berge cycle of length $\ell$ in a hypergraph $\mathcal{H}$ is a sequence of alternating vertices and edges $v_0e_0v_1e_1...v_\ell e_\ell v_0$ such that $\{v_i,v_{i+1}\}\subseteq e_i$ for all $i$, with indices taken modulo $\ell$. For $n$…

We study the structure of $r$-uniform hypergraphs containing no Berge cycles of length at least $k$ for $k \leq r$, and determine that such hypergraphs have some special substructure. In particular we determine the extremal number of such…

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…

For two graphs $F$ and $H$, the relative Tur\'{a}n number $\mathrm{ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau \cite{FKP} and Perarnau and Reed \cite{PR} studied these…