Related papers: An improved hypergraph Mantel's Theorem

Given a family of $3$-graphs $\mathcal{F}$, the uniform Tur\'{a}n density $\pi_{\therefore}(\mathcal{F})$ is defined as the infimum $d\in[0,1]$ for which any sufficiently large uniformly $d$-dense $3$-graph - that is, a $3$-graph which has…

The Tur\'an number of an r-uniform hypergraph H is the maximum number of edges in any r-graph on n vertices which does not contain H as a subgraph. Let P_l^(r) denote the family of r-uniform loose paths on l edges, F(k,l) denote the family…

For every positive integer $t$ we construct a finite family of triple systems ${\mathcal M}_t$, determine its Tur\'{a}n number, and show that there are $t$ extremal ${\mathcal M}_t$-free configurations that are far from each other in…

For integers $q\ge p\ge r\ge2$, we say that an $r$-uniform hypergraph $H$ has property $(q,p)$, if for any $q$-vertex subset $Q$ of $V(H)$, there exists a $p$-vertex subset $P$ of $Q$ spanning a clique in $H$. Let $T_{r}(n,q,p)=\min\{ e(H):…

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique…

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…

A fundamental result in extremal graph theory is attributed to Mantel's theorem, which states that every graph on $n$ vertices with more than $\lfloor n^2/4 \rfloor$ edges must contain a triangle. Lov\'{a}sz and Simonovits (1975) provided a…

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. In this paper we determine…

Let $\mathcal{F}$ be a family of $r$-graphs. The Tur\'an number $ex_r(n;\mathcal{F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{F}$-free. The famous Erd\H{o}s Matching Conjecture shows that…

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…

P. Erd\H{o}s [On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs $F$ that have to appear in any sufficiently large hypergraph $H$ of positive density. We…

For a graph $G$, denote by $t(G)$ (resp. $b(G)$) the maximum size of a triangle-free (resp. bipartite) subgraph of $G$. Of course $t(G) \geq b(G)$ for any $G$, and a classic result of Mantel from 1907 (the first case of Tur\'an's Theorem)…

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…

In this paper we describe a number of extensions to Razborov's semidefinite flag algebra method. We will begin by showing how to apply the method to significantly improve the upper bounds of edge and vertex Tur\'an density type results for…

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…

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…

A hypergraph is linear if any two edges intersect in at most one vertex. For a fixed $k$-uniform family ${\cal{F}}$ of hypergraphs, the linear Tur\'an number ${\rm ex}_{\rm lin}(n,{\cal{F}})$ is the maximum number of edges in a $k$-uniform…

For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by {ex}$(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not containing $F$ as a subgraph. Determining {ex}$(K_n,F)$ for a given graph $F$ is a…

The $r$-expansion $G^+$ of a graph $G$ is the $r$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a vertex subset of size $r-2$ disjoint from $V(G)$ such that distinct edges are enlarged by disjoint subsets. Let…

An $r$-uniform hypergraph is called an $r$-graph. A hypergraph is linear if every two edges intersect in at most one vertex. Given a linear $r$-graph $H$ and a positive integer $n$, the linear Tur\'an number $ex_L(n,H)$ is the maximum…