Related papers: Linear trees in uniform hypergraphs

Given an $r_0$-uniform hypergraph $F$, we define its $r$-uniform expansion $F^{(r)}$ to be the hypergraph obtained from $F$ by inserting $r-r_0$ distinct vertices into each edge of $F$, and we define $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ to be…

An $r$-uniform hypergraph is a tight $r$-tree if its edges can be ordered so that every edge $e$ contains a vertex $v$ that does not belong to any preceding edge and the set $e-v$ lies in some preceding edge. A conjecture of Kalai [Kalai],…

A hypergraph $H$ is said to be \emph{linear} if every pair of vertices lies in at most one hyperedge. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs, an $r$-uniform hypergraph $H$ is \emph{$\mathcal{F}$-free} if it contains no…

A connected and acyclic hypergraph is called a supertree. In this paper we mainly focus on the spectral radii of uniform supertrees. Li, Shao and Qi determined the first two $k$-uniform supertrees with large spectral radii among all the…

For all integers $k,d$ such that $k \geq 3$ and $k/2\leq d \leq k-1$, let $n$ be a sufficiently large integer {\rm(}which may not be divisible by $k${\rm)} and let $s\le \lfloor n/k\rfloor-1$. We show that if $H$ is a $k$-uniform hypergraph…

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…

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…

There has been interest recently in maximizing the number of independent sets in graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. Similarly, it is a corollary of the…

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…

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…

The transversal number $\tau(H)$ of a hypergraph $H$ is the minimum number of vertices that intersect every edge of $H$. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A $k$-uniform hypergraph…

The generalized Tur\'{a}n number $ex(n,K_s,H)$ is defined to be the maximum number of copies of a complete graph $K_s$ in any $H$-free graph on $n$ vertices. Let $F$ be a linear forest consisting of $k$ paths of orders…

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…

We prove an asymptotic result on the maximum number of k-vertex subtrees in binary trees of given order. This problem turns out to be equivalent to determine the maximum number of k+2-cycles in n-vertex outerplanar graphs, thus we settle…

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathscr{F}$ as a subgraph. The Tur\'an number, denoted by $ex(n, \mathscr{F})$, is the maximum number of edges in an $n$-vertex…

Let $F$ be a graph. A hypergraph is called Berge-$F$ if it can be obtained by replacing each edge of $F$ by a hyperedge containing it. Let $\mathcal{F}$ be a family of graphs. The Tur\'an number of Berge-$\mathcal{F}$ is the maximum…

The generalized Tur\'{a}n number $ex(n,K_s,H)$ is the maximum number of complete graph $K_s$ in an $H$-free graph on $n$ vertices. Let $F_k$ be the friendship graph consisting of $k$ triangles. Erd\H{o}s and S\'os (1976) determined the…

We study off-diagonal Ramsey numbers $r(H, K_n^{(k)})$ of $k$-uniform hypergraphs, where $H$ is a fixed linear $k$-uniform hypergraph and $K_n^{(k)}$ is complete on $n$ vertices. Recently, Conlon et al.\ disproved the folklore conjecture…

A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi shows that every large $n$-vertex graph with minimum degree at least $(1/2+\gamma)n$ contains all spanning trees of bounded degree. We generalised this result to loose spanning…

A tight $r$-tree $T$ is an $r$-uniform hypergraph that has an edge-ordering $e_1, e_2, \dots, e_t$ such that for each $i\geq 2$, $e_i$ has a vertex $v_i$ that does not belong to any previous edge and $e_i-v_i$ is contained in $e_j$ for some…