Related papers: When entropy meets Tur\'an: new proofs and hypergr…

Chao and Yu introduced an entropy method for hypergraph Tur\'an problems, and used it to show that the family of $\lfloor k/2\rfloor$ $k$-uniform tents have Tur\'an density $k!/k^k$. Il'kovi\v{c} and Yan improved this by reducing to a…

In a recent paper, Chao and Yu used an entropy method to show that the Tur\'an density of a certain family $\mathcal{F}$ of $\lfloor r/2\rfloor$ triangle-like $r$-uniform hypergraphs is $r!/r^r$. Later, Liu determined for large $n$ the…

The present paper is concerned with the various algebraic structures supported by the set of Tur\'an densities. We prove that the set of Tur\'an densities of finite families of r-graphs is a non-trivial commutative semigroup, and as a…

In 1965, Motzkin and Straus established a remarkable connection between the order of a maximum clique and the Lagrangian of a graph and provided a new proof of Tur\'an's theorem using the connection. The connection of Lagrangians and…

We study extensions of Tur\'an Theorem in edge-weighted settings. A particular case of interest is when constraints on the weight of an edge come from the order of the largest clique containing it. These problems are motivated by…

The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $r-2$ new vertices. Keevash and Sidorenko~ have…

The Tur\'an density of an $r$-uniform hypergraph $\mathcal{H}$, denoted $\pi(\mathcal{H})$, is the limit of the maximum density of an $n$-vertex $r$-uniform hypergraph not containing a copy of $\mathcal{H}$, as $n \to \infty$. Denote by…

For a fixed positive integer $n$ and an $r$-uniform hypergraph $H$, the Tur\'an number $ex(n,H)$ is the maximum number of edges in an $H$-free $r$-uniform hypergraph on $n$ vertices, and the Lagrangian density of $H$ is defined as…

We study the codegree Tur\'an density of $\mathcal{C}_\ell^r$, the $r$-uniform hypergraph tight cycle of length $\ell$. A result of Han, Lo, and Sanhueza-Matamala states that if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is even,…

If $\mathcal{F}$ is a family of graphs then the Tur\'an density of $\mathcal{F}$ is determined by the minimum chromatic number of the members of $\mathcal{F}$. The situation for Tur\'an densities of 3-graphs is far more complex and still…

Denote by $\mathcal{C}^-_{\ell}$ the $3$-uniform hypergraph obtained by removing one hyperedge from the tight cycle on $\ell$ vertices. It is conjectured that the Tur\'an density of $\mathcal{C}^-_{5}$ is $1/4$. In this paper, we make…

The Tur\'an hypergraph problem asks to find the maximum number of $r$-edges in a $r$-uniform hypergraph on $n$ vertices that does not contain a clique of size $a$. When $r=2$, i.e., for graphs, the answer is well-known and can be found in…

Unlike graphs, determining Tur\'{a}n densities of hypergraphs is known to be notoriously hard in general. The essential reason is that for many classical families of $r$-uniform hypergraphs $\mathcal{F}$, there are perhaps many…

The study of uniform Tur\'an densities was initiated in the 1980s by Erd\H{o}s and S\'os. Given a $3$-graph $F$, the uniform Tur\'an density of $F$, $\pi_{\therefore}(F)$, is defined as the infimum $d\in[0,1]$ such that every $3$-graph $H$…

Grosu [On the algebraic and topological structure of the set of Tur\'{a}n densities. \emph{J. Combin. Theory Ser. B} \textbf{118} (2016) 137--185] asked if there exist an integer $r\ge 3$ and a finite family of $r$-graphs whose Tur\'{a}n…

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,…

We make some conjectures about extremal densities of daisy-free families, where a `daisy' is a certain hypergraph. These questions turn out to be related to some Turan problems in the hypercube, but they are also natural in their own right.

We show that the set $\Pi^{(k)}$ of Tur\'an densities of $k$-uniform hypergraphs has infinitely many accumulation points in $[0,1)$ for every $k \geq 3$. This extends an earlier result of ours showing that $\Pi^{(k)}$ has at least one 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…

A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set…