Related papers: On partial Steiner $(n,r,\ell)$-system process

A spanning tree $T$ of graph $G$ is a $\rho$-approximate universal Steiner tree (UST) for root vertex $r$ if, for any subset of vertices $S$ containing $r$, the cost of the minimal subgraph of $T$ connecting $S$ is within a $\rho$ factor of…

Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph the vertex cover number is at most $r-1$ times the matching number. This conjecture is only known to be true for $r\leq 3$. For intersecting hypergraphs, Ryser's…

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…

For a $k$-uniform hypergraph $F$ and a positive integer $n$, the Ramsey number $r(F,n)$ denotes the minimum $N$ such that every $N$-vertex $F$-free $k$-uniform hypergraph contains an independent set of $n$ vertices. A hypergraph is…

Given $k\ge3$ and $1\leq \ell< k$, an $(\ell,k)$-cycle is one in which consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which do not contain…

Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and R\"{o}dl conjectured that for any given $\mu>0$, there exists $n_0$ such that the following holds. Every $n$-vertex Steiner triple system…

We use the polynomial method of Guth and Katz to establish stronger and {\it more efficient} regularity and density theorems for such $k$-uniform hypergraphs $H=(P,E)$, where $P$ is a finite point set in ${\mathbb R}^d$, and the edge set…

A graph $G$ percolates in the $K_{r,s}$-bootstrap process if we can add all missing edges of $G$ in some order such that each edge creates a new copy of $K_{r,s}$, where $K_{r,s}$ is the complete bipartite graph. We study…

The hypergraph unreliability problem asks for the probability that a hypergraph gets disconnected when every hyperedge fails independently with a given probability. For graphs, the unreliability problem has been studied over many decades,…

We deal with an extremal problem concerning panchromatic colorings of hypergraphs. A vertex $r$-coloring of a hypergraph $H$ is \emph{panchromatic} if every edge meets every color. We prove that for every $3<r\leq\sqrt[3]{n/(100\ln n)}$,…

An ${\ell}$-cycle system ${\mathcal F}$ of a graph $\Gamma$ is a set of ${\ell}$-cycles which partition the edge set of $\Gamma$. Two such cycle systems ${\mathcal F}$ and ${\mathcal F}'$ are said to be {\em orthogonal} if no two distinct…

We propose a new approach to studies on partial Steiner triple systems consisting in determining complete graphs contained in them. We establish the structure which complete graphs yield in a minimal PSTS that contains them. As a by-product…

An $r$-uniform \textit{linear cycle} of length $\ell$, denoted by $C_{\ell}^r$, is an $r$-graph with edges $e_1, \ldots, e_{\ell}$ such that for every $i\in [\ell-1]$, $|e_i\cap e_{i+1}|=1$, $|e_{\ell}\cap e_1|=1$ and $e_i\cap…

A transversal coalition in a hypergraph $H$ is a partition of the vertex set $U$ into two subsets $U_1$ and $U_2$ such that neither $U_1$ nor $U_2$ alone intersects every hyperedge of $H$, but their union, $U_1 \cup U_2$, intersects every…

Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation is a full triangulation of some subset P' of P containing all extreme points in…

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is…

We say that a tree $T$ is an $S$-Steiner tree if $S \subseteq V(T)$ and a hypergraph is an $S$-Steiner hypertree if it can be trimmed to an $S$-Steiner tree. We prove that it is NP-complete to decide, given a hypergraph $\mathcal{H}$ and…

In 1973, Erd\H{o}s conjectured the existence of high girth $(n,3,2)$-Steiner systems. Recently, Glock, K\"{u}hn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erd\H{o}s' conjecture. Just this year,…

We propose an efficient algorithm for matching two correlated Erd\H{o}s--R\'enyi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence. When the edge density $q= n^{- \alpha+o(1)}$ for a constant $\alpha…

Two hypergraphs $H_1,\ H_2$ are called {\em cross-intersecting} if $e_1 \cap e_2 \neq \emptyset$ for every pair of edges $e_1 \in H_1,~e_2 \in H_2$. Each of the hypergraphs is then said to {\em block} the other. Given parameters $n,r,m$ we…