Related papers: On Ryser's Conjecture for Linear Intersecting Mult…

A subset $M$ of the edges of a graph or hypergraph is hitting if $M$ covers each vertex of $H$ at least once, and $M$ is $t$-shallow if it covers each vertex of $H$ at most $t$ times. We consider the existence of shallow hitting edge sets…

A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. We prove a lower bound of $\frac{2}{3}n-1$, improving on the trivial $\frac{1}{2}n$, and an analogous result for…

Determine the size of $r$-graphs with given graph parameters is an interesting problem. Chv\'atal and Hanson (JCTB, 1976) gave a tight upper bound of the size of 2-graphs with restricted maximum degree and matching number; Khare (DM, 2014)…

Suppose that $T$ is an acyclic $r$-uniform hypergraph, with $r\ge 2$. We define the ($t$-color) chromatic Ramsey number $\chi(T,t)$ as the smallest $m$ with the following property: if the edges of any $m$-chromatic $r$-uniform hypergraph…

A set $A$ of vertices in an $r$-uniform hypergraph $\mathcal H$ is covered in $\mathcal H$ if there is some vertex $u\not\in A$ such that, for every $(r-1)$-set $B\subset A$, the set $\{u\}\cup B$ is in $\mathcal H$. Erdos and Moser (1970)…

In their famous 1974 paper introducing the local lemma, Erd\H{o}s and Lov\'asz posed a question-later referred by Erd\H{o}s as one of his three favorite open problems: What is the minimum number of edges in an $r$-uniform, intersecting…

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…

For a graph G and integer r \geq 1 we denote the family of independent r-sets of V(G) by I^{(r)}(G). A graph G is said to be r-EKR if no intersecting subfamily of I^{(r)}(G) is larger than the largest such family all of whose members…

We consider drawings of graphs that contain dense subgraphs. We introduce intersection-link representations for such graphs, in which each vertex $u$ is represented by a geometric object $R(u)$ and in which each edge $(u,v)$ is represented…

Let $f_r(n,v,e)$ denote the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices, in which the union of any $e$ distinct edges contains at least $v+1$ vertices. The study of $f_r(n,v,e)$ was initiated by Brown, Erd{\H{o}}s…

Let $n,s,$ and $k$ be positive integers such that $k\geq 3$, $s\geq 3$ and $n\geq ks$. An $s$-matching $M_s$ in a $k$-uniform hypergraph is a set of $s$ pairwise disjoint edges. The anti-Ramsey number $\textrm{ar}(n,k,M_s)$ of an…

For $k \ge 2$, let $H$ be a $k$-uniform hypergraph on $n$ vertices and $m$ edges. The transversal number $\tau(H)$ of $H$ is the minimum number of vertices that intersect every edge. Chv\'{a}tal and McDiarmid [Combinatorica 12 (1992),…

More than forty years ago, Erd\H{o}s conjectured that for any T <= N/K, every K-uniform hypergraph on N vertices without T disjoint edges has at most max{\binom{KT-1}{K}, \binom{N}{K} - \binom{N-T+1}{K}} edges. Although this appears to be a…

Let $\mathcal{H}=(V,\mathcal{E})$ be an $r$-uniform hypergraph on $n$ vertices and fix a positive integer $k$ such that $1\le k\le r$. A $k$-\emph{matching} of $\mathcal{H}$ is a collection of edges $\mathcal{M}\subset \mathcal{E}$ such…

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all…

We prove the well-known Brown-Erd\H{o}s-S\'os Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is…

For $r \ge 2$, an $r$-uniform hypergraph is called a friendship $r$-hypergraph if every set $R$ of $r$ vertices has a unique 'friend' - that is, there exists a unique vertex $x \notin R$ with the property that for each subset $A \subseteq…

Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B, 141 (2020), 343-351] asked whether every $r$-edge-connected $r$-regular graph of even order has $r-2$ pairwise disjoint perfect matchings. We show that this is…

In this paper, we study the problem of determining the maximum number of edges in an $n$-vertex $r$-uniform hypergraph that contains no $(k+1)$-connected subgraph. The graph case is a classical problem initiated by Mader, central to graph…

The vertices of any graph with $m$ edges may be partitioned into two parts so that each part meets at least $\frac{2m}{3}$ edges. Bollob\'as and Thomason conjectured that the vertices of any $r$-uniform hypergraph with $m$ edges may…