Related papers: New Tur\'an densities for 3-graphs

A fundamental barrier in extremal hypergraph theory is the presence of many near-extremal constructions with very different structures. Indeed, the classical constructions due to Kostochka imply that the notorious extremal problem for the…

We identify three 3-graphs on five vertices each missing in all known extremal configurations for Turan's (3,4)-problem and prove Turan's conjecture for 3-graphs that are additionally known not to contain any induced copies of these…

The \emph{Ramsey multiplicity constant} of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labeled copies of $H$ in a $2$-edge colouring of $K_n$. Fox and Wigderson recently identified a large…

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,…

For fixed integers $r>k\ge 2,e\ge 3$, let $f_r(n,er-(e-1)k,e)$ be the maximum number of edges in an $r$-uniform hypergraph in which the union of any $e$ distinct edges contains at least $er-(e-1)k+1$ vertices. A classical result of Brown,…

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 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…

We prove that, for every integer $r\ge 3$, the set $\Pi^{(r)}_\infty$ of Tur\'an densities of (possibly infinite) families of $r$-graphs contains non-degenerate intervals, including an interval of the form $[1-\delta_r,1]$ for some…

An $r$-uniform graph $G$ is dense if and only if every proper subgraph $G'$ of $G$ satisfies $\lambda (G') < \lambda (G)$, where $\lambda (G)$ is the Lagrangian of a hypergraph $G$. In 1980's, Sidorenko showed that $\pi(F)$, the Tur\'an…

We introduce a modification of the Tur\'an density of ordered graphs and investigate this graph parameter.

Given a family $\mathcal{F}$ of $r$-graphs, the Tur\'{a}n number of $\mathcal{F}$ for a given positive integer $N$, denoted by $ex(N,\mathcal{F})$, is the maximum number of edges of an $r$-graph on $N$ vertices that does not contain any…

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…

Let $I(F,n)$ denote the maximum number of induced copies of a graph $F$ in an $n$-vertex graph. The inducibility of $F$, defined as $i(F)=\lim_{n\to \infty} I(F,n)/\binom{n}{v(F)}$, is a central problem in extremal graph theory. In this…

The $3$-uniform tight $\ell$-cycle $C_\ell^{3}$ is the $3$-graph on $\{1,\dots,\ell\}$ consisting of all $\ell$ consecutive triples in the cyclic order. Let $\mathcal{C}$ be either the pair $\{C_{4}^{3}, C_{5}^{3}\}$ or the single tight…

Fix a graph $F$. We say that a graph is {\it $F$-free} if it does not contain $F$ as a subgraph. The {\it Tur\'an number} of $F$, denoted $\mathrm{ex}(n,F)$, is the maximum number of edges possible in an $n$-vertex $F$-free graph. The study…

Given a $k$-graph $H$ a complete blow-up of $H$ is a $k$-graph $\hat{H}$ formed by replacing each $v\in V(H)$ by a non-empty vertex class $A_v$ and then inserting all edges between any $k$ vertex classes corresponding to an edge of $H$.…

We study a Tur\'an-type problem on edge-colored complete graphs. We show that for any $r$ and $t$, any sufficiently large $r$-edge-colored complete graph on $n$ vertices with $\Omega(n^{2-1/tr^r})$ edges in each color contains a member from…

Let $F_{3,3}$ be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. We show that for all $n \ge 6$, the maximum…

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…

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…