Related papers: On $3$-graphs with vanishing codegree Tur\'{a}n de…

For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Tur\'{a}n density $\pi_{\mathrm{co}}(F)$ is the infimum over all $\alpha$ such that any $n$-vertex $k$-graph $H$ with every $(k-1)$-subset of $V(H)$ contained in at least…

The codegree Tur\'an density $\pi_{\text{co}}(F)$ of a $k$-uniform hypergraph (or $k$-graph) $F$ is the infimum over all $d$ such that a copy of $F$ is contained in any sufficiently large $n$-vertex $k$-graph $G$ with the property that any…

The uniform Tur\'an density $\pi_{1}(F)$ of a $3$-uniform hypergraph $F$ is the supremum over all $d$ for which there is an $F$-free hypergraph with the property that every linearly sized subhypergraph with density at least $d$. Determining…

Let $k\geq 3$. Given a $k$-uniform hypergraph $H$, the minimum codegree $\delta(H)$ is the largest $d\in\mathbb{N}$ such that every $(k-1)$-set of $V(H)$ is contained in at least $d$ edges. Given a $k$-uniform hypergraph $F$, the codegree…

The uniform Tur\'an density $\pi_{u}(F)$ of a $3$-uniform hypergraph (or $3$-graph) $F$ is the supremum of all $d$ such that there exist infinitely many $F$-free $3$-graphs $H$ in which every induced subhypergraph on a linearly sized vertex…

Here we consider the hypergraph Tur\'an problem in uniformly dense hypergraphs as was suggested by Erd\H{o}s and S\'os. Given a $3$-graph $F$, the uniform Tur\'an density $\pi_u(F)$ of $F$ is defined as the supremum over all $d\in[0,1]$ for…

Let $H$ be a $k$-graph (i.e. a $k$-uniform hypergraph). Its minimum codegree $\delta_{k-1}(H)$ is the largest integer $t$ such that every $(k-1)$-subset of $V(H)$ is contained in at least $t$ edges of~$H$. The \emph{codegree Tur\'an…

Given an $r$-graph $H$ on $h$ vertices, and a family $\mathcal{F}$ of forbidden subgraphs, we define $\ex_{H}(n, \mathcal{F})$ to be the maximum number of induced copies of $H$ in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Then the…

For $k\ge 3$, the $(k-2)$-uniform Tur\'an density $\pi_{k-2}(F)$ of a $k$-graph $F$ is the supremum of $d$ for which there are arbitrarily large $F$-free $k$-graphs that are uniformly $d$-dense with respect to the $k$-vertex cliques of…

A $k$-graph (or $k$-uniform hypergraph) $H$ is uniformly dense if the edge distribution of $H$ is uniformly dense with respect to every large collection of $k$-vertex cliques induced by sets of $(k-2)$-tuples. Reiher, R\"odl and Schacht…

For integers $1\le \ell<k$, the $\ell$-degree Tur\'an density $\pi_\ell(F)$ measures the minimum $\ell$-degree threshold that forces a copy of a fixed $k$-uniform hypergraph $F$, generalizing both the classical Tur\'an density $\pi_1$ and…

The \emph{minimum positive co-degree} of a non-empty $r$-graph ${H}$, denoted $\delta_{r-1}^+( {H})$, is the maximum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $ {H}$, then $S$ is contained in at least $k$ distinct…

For an ordered graph $F$, denote the Tur\'an density by $\vec{\pi}(F)$. The relative Tur\'an density, denoted by $\rho(F)$, is the supremum over $\alpha \in [0,1]$ such that every ordered graph $G$ contains an $F$-free subgraph $G'$ with…

A $3$-uniform hypergraph (or $3$-graph) $H=(V,E)$ is $(d,\mu,1)$-\emph{dense} if for any subsets $X,Y,Z\subseteq V$, the number of triples $(x,y,z)\in X\times Y\times Z$ such that $\{x,y,z\}$ is an edge of $H$ is at least $d|X||Y||Z|-\mu…

In the 1980s, Erd\H{o}s and S\'os first introduced an extremal problem on hypergraphs with density constraints. Given an $r$-uniform hypergraph $F$ (or $r$-graph for short), its uniform Tur\'an density $\pi_u(F)$ is the smallest value of…

Let $r\ge 3$. Given an $r$-graph $H$, the minimum codegree $\delta_{r-1}(H)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(H)$ is contained in at least $t$ edges of $H$. Given an $r$-graph $F$, the codegree Tur\'an density…

The {\em Tur\'an number} of an $r$-uniform graph $F$, denoted by $ex(n,F)$, is the maximum number of edges in an $F$-free $r$-uniform graph on $n$ vertices. The {\em Tur\'{a}n density} of $F$ is defined as…

The codegree Tur\'an density $\gamma(F)$ of a $k$-graph $F$ is the smallest $\gamma\in[0,1)$ such that every $k$-graph $H$ with $\delta_{k-1}(H)\geq(\gamma+o(1))\vert V(H)\vert$ contains a copy of $F$. We prove that for all…

Write $K^{(k)}_{n}$ for the complete $k$-graph on $n$ vertices. For $2 \leq k \leq g < r$ integers, let $\pi\left(n, K^{(k)}_{g}, K^{(k)}_r\right)$ be the maximum density of $K^{(k)}_{g}$ in $n$ vertex $K^{(k)}_{r}$-free $k$-graphs. The…

The codegree Tur\'an density $\gamma(F)$ of a $k$-uniform hypergraph $F$ is the minimum real number $\gamma \ge 0$ such that every $k$-uniform hypergraph on sufficiently many $n$ vertices, in which every set of $k-1$ vertices is contained…