A well-known conjecture, often attributed to Ryser, states that the cover number of an $r$-partite $r$-uniform hypergraph is at most $r - 1$ times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Király and Tóthmérész considered the problem under the assumption that the hypergraph is $t$-intersecting, conjecturing that the cover number $τ(\mathcal{H})$ of such a hypergraph $\mathcal{H}$ is at most $r - t$. In these papers, it was proven that the conjecture is true for $r \leq 4t-1$, but also that it need not be sharp; when $r = 5$ and $t = 2$, one has $τ(\mathcal{H}) \leq 2$.
We extend these results in two directions. First, for all $t \geq 2$ and $r \leq 3t-1$, we prove a tight upper bound on the cover number of these hypergraphs, showing that they in fact satisfy $τ(\mathcal{H}) \leq \lfloor(r - t)/2 \rfloor + 1$. Second, we extend the range of $t$ for which the conjecture is known to be true, showing that it holds for all $r \leq \frac{36}{7}t-5$. We also introduce several related variations on this theme. As a consequence of our tight bounds, we resolve the problem for $k$-wise $t$-intersecting hypergraphs, for all $k \geq 3$ and $t \geq 1$. We further give bounds on the cover numbers of strictly $t$-intersecting hypergraphs and the $s$-cover numbers of $t$-intersecting hypergraphs.
A famous conjecture of Ryser states that every $r$-partite hypergraph has vertex cover number at most $r - 1$ times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as $r$-Ryser hypergraphs, have been studied extensively. It was recently proved by Haxell, Narins and Szabó that all $3$-Ryser hypergraphs with matching number $ν> 1$ are essentially obtained by taking $ν$ disjoint copies of intersecting $3$-Ryser hypergraphs. Abu-Khazneh showed that such a characterisation is false for $r = 4$ by giving a computer generated example of a $4$-Ryser hypergraph w…
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 $r\leq 3$ in general and for $r\leq 5$ if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. $r$-partite hypergraphs whose cover number is $r-1$ times its matching number. Aside from a few sporadic examples, the set of uniformities $r$ for which Ryser's Conjecture is known to be tight is limited to those integer…
Ryser conjectured that $τ\le(r-1)ν$ for $r$-partite hypergraphs, where $τ$ is the covering number and $ν$ is the matching number. We prove this conjecture for $r\le9$ in the special case of linear intersecting hypergraphs, in other words where every pair of lines meets in exactly one vertex. Aharoni formulated a stronger version of Ryser's conjecture which specified that each $r$-partite hypergraph should have a cover of size $(r-1)ν$ of a particular form. We provide a counterexample to Aharoni's conjecture with $r=13$ and $ν=1$. We also report a number of computational results. For $r=7$, we …
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 Conjecture reduces to saying that the edges of every $r$-partite intersecting hypergraph can be covered by $r-1$ vertices. This special case of the conjecture has only been proven for $r \leq 5$. It is interesting to study hypergraphs which are extremal in Ryser's Conjecture i.e, those hypergraphs for which the vertex cover number is exactly $r-1$ times the matc…
A famous conjecture of Ryser states that any $r$-partite set system has transversal number at most $r-1$ times their matching number. This conjecture is only known to be true for $r\leq3$ in general, for $r\leq5$ if the set system is intersecting, and for $r\leq9$ if the intersecting set system is linear. In this note, we deal with Ryser's Conjecture for intersecting $r$-partite linear systems; that is, if $τ$ is the transversal number for an intersecting $r$-partite linear system, then Ryser's Conjecture states that $τ\leq r-1$. If this conjecture is true, this is known to be sharp for $r$ fo…
Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $r - 1$ times the size of the largest matching. For $r = 2$, the conjecture is simply König's Theorem and every bipartite graph is a witness for its tightness. The conjecture has also been proven for $r = 3$ by Aharoni using topological methods, but the proof does not give information on the extremal $3$-uniform hypergraphs. Our goal in this paper is to characterize those hypergraphs which are tight for Aharoni's Theorem. Our proof of this characterization is also based on topological machi…
Let $τ(\mathcal{H})$ be the cover number and $ν(\mathcal{H})$ be the matching number of a hypergraph $\mathcal{H}$. Ryser conjectured that every $r$-partite hypergraph $\mathcal{H}$ satisfies the inequality $τ(\mathcal{H}) \leq (r-1) ν(\mathcal{H})$. This conjecture is open for all $r \ge 4$. For intersecting hypergraphs, namely those with $ν(\mathcal{H})=1$, Ryser's conjecture reduces to $τ(\mathcal{H}) \leq r-1$. Even this conjecture is extremely difficult and is open for all $ r \ge 6$. For infinitely many $r$ there are examples of intersecting $r$-partite hypergraphs with $τ(\mathcal{H})=r…
A famous conjecture (usually called Ryser's conjecture) that appeared in the Ph.D thesis of his student, J.~R.~Henderson [15], states that for an $r$-uniform $r$-partite hypergraph $\mathcal{H}$, the inequality $τ(\mathcal{H})\le(r-1)\cdot ν(\mathcal{H})$ always holds. This conjecture is widely open, except in the case of $r=2$, when it is equivalent to K\H onig's theorem [18], and in the case of $r=3$, which was proved by Aharoni in 2001 [3]. Here we study some special cases of Ryser's conjecture. First of all the most studied special case is when $\mathcal{H}$ is intersecting. Even for this …
A famous conjecture of Ryser is that in an $r$-partite hypergraph the covering number is at most $r-1$ times the matching number. If true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has covering number at least $r-1$. We find that $f(r)$ is achieved only by linear hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}…
In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the lower bound $\frac{\dim(\mathcal{I}(G))}{2} - 2$ on the connectedness of the independence complex $\mathcal{I}(G)$ of line graphs of bipartite graphs $G$ is tight. In our main theorem we characterize the extremal examples. Our proof of this characterization is based on topological machinery. Our motivation for studying this problem comes from a classical conje…
Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $ν(H)$, the vertex cover number is at most $(r-1)ν(H)$. This far reaching generalization of König's theorem is only known to be true for $r\leq 3$, or $ν(G)=1$ and $r\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $α(G)$, there exists at most $(r-1)α(G)$ monochromatic connected subgraphs which cover the vertex set of $G$. We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stron…
We give a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r-4 for all but finitely many r. This answers a question of Abu-Khazneh, Barat, Pokrovskiy and Szabo, and shows that a long-standing unsolved conjecture due to Ryser is close to being best possible for every value of r.
A matching in a hypergraph $H$ is a set of pairwise vertex disjoint edges in $H$ and the matching number of $H$ is the maximum cardinality of a matching in $H$. A transversal in $H$ is a subset of vertices in $H$ that has a nonempty intersection with every edge of $H$. The transversal number $τ(H)$ of $H$ is the minimum cardinality of a transversal in $H$. A hypergraph $H$ is an intersecting hypergraph if every two distinct edges of $H$ have a non-empty intersection. Equivalently, $H$ is an intersecting hypergraph if and only if it has matching number one. In this paper we study the extremal b…
Using the Filmus-Golubev-Lifshitz method to bound the independence number of a hypergraph, we solve some problems concerning multiply intersecting families with biased measure. Among other results we obtain a stability result of a measure version of the Erdos-Ko-Rado theorem for multiply intersecting families.
Two prominent conjectures by Herbert J. Ryser have been falsely attributed to a somewhat obscure conference proceedings that he wrote in German. Here we provide a translation of that paper and try to correct the historical record at least as far as what was conjectured in it. The two conjectures relate to transversals in Latin squares of odd order and to the relationship between the covering number and the matching number of multipartite hypergraphs.
A hypergraph is 2-intersecting if any two edges intersect in at least two vertices. Blais, Weinstein and Yoshida asked (as a first step to a more general problem) whether every 2-intersecting hypergraph has a vertex coloring with a constant number of colors so that each hyperedge e has at least min{|e|,3} colors. We show that there is such a coloring with at most 5 colors (which is best possible).
A rainbow matching for (not necessarily distinct) sets F_1,...,F_k of hypergraph edges is a matching consisting of k edges, one from each F_i. The aim of the paper is twofold - to put order in the multitude of conjectures that relate to this concept (some of them first presented here), and to present some partial results on one of these conjectures, that seems central among them.
Let $c$ be a positive constant. We show that if $r=\lfloor cn^{1/3}\rfloor$ and the members of ${[n]\choose r}$ are chosen sequentially at random to form an intersecting hypergraph then with limiting probability $(1+c^3)^{-1}$, as $n\to\infty$, the resulting family will be of maximum size ${n-1\choose r-1}$.
A matching in a hypergraph $H$ is a set of pairwise disjoint hyperedges. The matching number $α'(H)$ of $H$ is the size of a maximum matching in $H$. A subset $D$ of vertices of $H$ is a dominating set of $H$ if for every $v\in V\setminus D$ there exists $u\in D$ such that $u$ and $v$ lie in an hyperedge of $H$. The cardinality of a minimum dominating set of $H$ is called the domination number of $H$, denoted by $γ(H)$. It is known that for a intersecting hypergraph $H$ with rank $r$, $γ(H)\leq r-1$. In this paper we present structural properties on intersecting hypergraphs with rank $r$ satis…
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 determine the maximal size of a sub-hypergraph of $[n]^r$ (meaning that it is $r$-partite, with all sides of size $n$) for which there exists a blocking sub-hypergraph of $[n]^r$ of size $m$. The answer involves a fractal-like (that is, self-similar) sequence, first studied by Knuth. We also study the same question with $\binom{n}{r}$ replacing $[n]^r$.
The celebrated Erdős-Ko-Rado theorem shows that for $n \ge 2k$ the largest intersecting $k$-uniform set family on $[n]$ has size $\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set families must be. Katona, Katona and Katona introduced the notion of most probably intersecting families, which maximise the probability of random subfamilies being intersecting. We study the most probably intersecting problem for $k$-uniform set families. We provide a rough structural characterisation of the most probably intersecting families and, for families of particular sizes, show th…
For any c >= 2, a c-strong coloring of the hypergraph G is an assignment of colors to the vertices of G such that for every edge e of G, the vertices of e are colored by at least min{c,|e|} distinct colors. The hypergraph G is t-intersecting if every two edges of G have at least t vertices in common. We ask: for fixed c >= 2 and t >= 1, what is the minimum number of colors that is sufficient to c-strong color any t-intersecting hypergraphs? The purpose of this note is to answer the question for some values of t and c and, more importantly, to describe the settings for which the question is sti…
Let $c$ be a positive constant. Suppose that $r=o(n^{5/12})$ and the members of $\binom{[n]}{r}$ are chosen sequentially at random to form an intersecting hypergraph $\mathcal{H}$. We show that whp $\mathcal{H}$ consists of a simple hypergraph $\mathcal{S}$ of size $Θ(r/n^{1/3})$, a distinguished vertex $v$ and all $r$-sets which contain $v$ and meet every edge of $\mathcal{S}$. This is a continuation of the study of such random intersecting systems started in [Electron. J. Combin, (2003) R29] where the case $r=O(n^{1/3})$ was considered. To obtain the stated result we continue to investigate …
Erdős and Lovász noticed that an $r$-uniform intersecting hypergraph $H$ with maximal covering number, that is $τ(H)=r$, must have at least $\frac{8}{3}r-3$ edges. There has been no improvement on this lower bound for 45 years. We try to understand the reason by studying some small cases to see whether the truth lies very close to this simple bound. Let $q(r)$ denote the minimum number of edges in an intersecting $r$-uniform hypergraph. It was known that $q(3)=6$ and $q(4)=9$. We obtain the following new results: The extremal example for uniformity 4 is unique. Somewhat surprisingly it is not …
For a hypergraph $\mathcal{H}$, define the minimum positive codegree $δ_i^+(\mathcal{H})$ to be the largest integer $k$ such that every $i$-set which is contained in at least one edge of $\mathcal{H}$ is contained in at least $k$ edges. For $1\le s\le k,t$ and $t\le r$, we prove that for $n$-vertex $t$-intersecting $r$-graphs $\mathcal{H}$ with $δ_{r-s}^+(\mathcal{H})>{k-1\choose s}$, the unique hypergraph with the maximum number of edges is the hypergraph $\mathcal{H}$ consisting of every edge which intersects a set of size $2k-2s+t$ in at least $k-s+t$ vertices provided $n$ is sufficiently l…
A hypergraph $\mathcal{F}$ is non-trivial intersecting if every two edges in it have a nonempty intersection but no vertex is contained in all edges of $\mathcal{F}$. Mubayi and Verstraëte showed that for every $k \ge d+1 \ge 3$ and $n \ge (d+1)n/d$ every $k$-graph $\mathcal{H}$ on $n$ vertices without a non-trivial intersecting subgraph of size $d+1$ contains at most $\binom{n-1}{k-1}$ edges. They conjectured that the same conclusion holds for all $d \ge k \ge 4$ and sufficiently large $n$. We confirm their conjecture by proving a stronger statement. They also conjectured that for $m \ge 4$ a…
Given an integer $r$ and a vector $\vec{a}=(a_1, \ldots ,a_p)$ of positive numbers with $\sum_{i \le p} a_i=r$, an $r$-uniform hypergraph $H$ is said to be $\vec{a}$-partitioned if $V(H)=\bigcup_{i \le p}V_i$, where the sets $V_i$ are disjoint, and $|e \cap V_i|=a_i$ for all $e \in H,~~i \le p$. A $\vec{1}$-partitioned hypergraph is said to be $r$-partite. Let $t(\vec{a})$ be the maximum, over all intersecting $\vec{a}$-partitioned hypergraphs $H$, of the minimal size of a cover of $H$. A famous conjecture of Ryser is that $t(\vec{1})\le r-1$. Tuza conjectured that if $r>2$ then $t(\vec{a})=r$…
The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turan numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k in {3,4}. We give a complete solution when k=3 and an almost complete solution (with eleven exceptio…
For a hypergraph $H$, define its intersection spectrum $I(H)$ as the set of all intersection sizes $|E\cap F|$ of distinct edges $E,F\in E(H)$. In their seminal paper from 1973 which introduced the local lemma, Erdős and Lovász asked: how large must the intersection spectrum of a $k$-uniform $3$-chromatic intersecting hypergraph be? They showed that such a hypergraph must have at least three intersection sizes, and conjectured that the size of the intersection spectrum tends to infinity with $k$. Despite the problem being reiterated several times over the years by Erdős and other researchers, …
Let $\mc{K}_{3,3}^3$ be the 3-graph with 15 vertices $\{x_i, y_i: 1 \le i \le 3\}$ and $\{z_{ij}: 1 \le i,j \le 3\}$, and 11 edges $\{x_1, x_2, x_3\}$, $\{y_1, y_2, y_3\}$ and $\{\{x_i, y_j, z_{ij}\}: 1 \le i,j \le 3\}$. We show that for large $n$, the unique largest $\mc{K}_{3,3}^3$-free 3-graph on $n$ vertices is a balanced blow-up of the complete 3-graph on 5 vertices. Our proof uses the stability method and a result on lagrangians of intersecting families that has independent interest.
Intersecting and cross-intersecting families usually appear in extremal combinatorics in the vein of the Erd{\H o}s--Ko--Rado theorem. On the other hand, P.~Erd{\H o}s and L.~Lov{á}sz in the noted paper~\cite{EL} posed problems on coloring intersecting families as a restriction of classical hypergraph coloring problems to a special class of hypergraphs. This note deals with the mentioned coloring problems stated for cross-intersecting families.
A $k$-uniform hypergraph is $s$-almost intersecting if every edge is disjoint from exactly $s$ other edges. Gerbner, Lemons, Palmer, Patkós and Szécsi conjectured that for every $k$, and $s>s_0(k)$, every $k$-uniform $s$-almost intersecting hypergraph has at most $(s+1)\binom{2k-2}{k-1}$ edges. We prove a strengthened version of this conjecture and determine the extremal graphs. We also give some related results and conjectures.
Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{minimum positive co-degree} of $\mathcal{H}$, denoted by $δ_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $\mathcal{H}$, then $S$ is contained in at least $k$ hyperedges of $\mathcal{H}$. For $r\geq k$ fixed and $n$ sufficiently large, we determine the maximum possible size of an intersecting $r$-uniform $n$-vertex hypergraph with minimum positive co-degree $δ_{r-1}^+(\mathcal{H}) \geq k$ and characterize the unique hypergraph attaining this maximum. This generalizes the Erd\H o…
Erdős-Ko-Rado-type problem for hypergraph matchings
School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）；Hebei Provincial Key Laboratory of Mathematical Theory and Analysis for Network and Data Science, Beijing Jiaotong University（北京交通大学网络与数据科学数学理论与分析河北省重点实验室）；School of Mathematics and Statistics, Ningbo University（宁波大学数学与统计学院）；Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）
Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $\mathcal{F}$ of $\frac{n}2$-element subsets of $X$, one can partition $X$ into $\frac{n}2$ disjoint pairs in such a way that no matter how we pick one element from each of the first $\frac{n}2 - 1$ pairs, the set formed by them can always be completed to a member of $\mathcal{F}$ by adding an element of the last pair. The above problem is related to classical questions in extremal set theory. For any $t\ge 2$, we call a family of s…
Consider a graph $G$ and a $k$-uniform hypergraph $\mathcal{H}$ on common vertex set $[n]$. We say that $\mathcal{H}$ is $G$-intersecting if for every pair of edges in $X,Y \in \mathcal{H}$ there are vertices $x \in X$ and $y \in Y$ such that $x = y$ or $x$ and $y$ are joined by an edge in $G$. This notion was introduced by Bohman, Frieze, Ruszinkó and Thoma who proved a natural generalization of the Erdős-Ko-Rado Theorem for $G$-intersecting $k$-uniform hypergraphs for $G$ sparse and $k = O( n^{1/4} )$. In this note, we extend this result to $k = O\left( \sqrt{n} \right)$.
In 1973 P. Erdős and L. Lovász noticed that any hypergraph whose edges are pairwise intersecting has chromatic number 2 or 3. In the first case, such hypergraph may have any number of edges. However, Erdős and Lovász proved that in the second case, the number of edges is bounded from above. For example, if a hypergraph is $ n $-uniform, has pairwise intersecting edges, and has chromatic number 3, then the number of its edges does not exceed $ n^n $. Recently D.D. Cherkashin improved this bound (see \cite{Ch}). In this paper, we further improve it in the case when the number of vertices of an $…
For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called $t$-intersecting if $|F\cap F'|\geq t$ for all $F,F'\in \mathcal{F}$. One of the central results of extremal set theory is the Erdős-Ko-Rado Theorem which states that for $n\geq (k-t+1)(t+1)$ no $t$-intersecting $k$-graph has more than $\binom{n-t}{k-t}$ edges. For $n$ greater than this threshold the $t$-star (all $k$-sets containing a fixed $t$-set) is the only fa…
Let $\mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $\mathcal{F}\subset\mathcal{H}$ with covering number at most $p$ is $\left(\binom{r+p}{r}-1\right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $\mathcal{F}\subset\mathcal{H}$ is $\frac{1}{2}\binom{2r}{r}$-collapsible.
We investigate the asymptotic version of the Erdős-Ko-Rado theorem for the random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$. For $2 \leq k(n) \leq n/2$, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to 1 as $n\to\infty$, the largest intersecting subhypergraph of $\mathcal{H}^k(n,p)$ has size $(1+o(1))p\frac kn N$, for any $p\gg \frac nk\ln^2\!\left(\frac nk\right)D^{-1}$. This lower bound on $p$ is asymptotically best possible for $k=Θ(n)$. For this range of $k$ and $p$, we are able to show stability as well. A different behavior occurs when $k = o(n)$. In th…
A family of sets is intersecting if no two of its members are disjoint, and has the Erdős-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has nonempty intersection. Denote by $\mathcal{H}_k(n,p)$ the random family in which each $k$-subset of $\{1\dots n\}$ is present with probability $p$, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks: \[ \mbox{for what $p=p(n,k)$ is $\mathcal{H}_k(n,p)$ likely to be EKR?} \] Here, for fixed $c<1/4$, and $k< \sqrt{cn\log n}$ we give a precise answer to this question, characterizing those se…
The study of intersecting structures is central to extremal combinatorics. A family of permutations $\mathcal{F} \subset S_n$ is \emph{$t$-intersecting} if any two permutations in $\mathcal{F}$ agree on some $t$ indices, and is \emph{trivial} if all permutations in $\mathcal{F}$ agree on the same $t$ indices. A $k$-uniform hypergraph is \emph{$t$-intersecting} if any two of its edges have $t$ vertices in common, and \emph{trivial} if all its edges share the same $t$ vertices. The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that…
A linear hypergraph is intersecting if any two different edges have exactly one common vertex and an $n$-quasicluster is an intersecting linear hypergraph with $n$ edges each one containing at most $n$ vertices and every vertex is contained in at least two edges. The Erdös-Faber-Lovász Conjecture states that the chromatic number of any $n$-quasicluster is at most $n$. In the present note we prove the correctness of the conjecture for a new infinite class of $n$-quasiclusters using a specific edge coloring of the complete graph.
Let $S^r(n)$ be the $r$-graph on $n$ vertices with parts $A$ and $B$, where the edges consist of all $r$-tuples with $1$ vertex in $A$ and $r-1$ vertices in $B$, and the sizes of $A$ and $B$ are chosen to maximise the number of edges. Let $M_t^r$ be the $r$-graph with $t$ pairwise disjoint edges. Given an $r$-graph $F$ and a positive integer $p\geq |V(F)|$, we define the {\em extension} of $F$, denoted by $H_{p}^{F}$ as follows: Label the vertices of $F$ as $v_1,\dots,v_{|V(F)|}$. Add new vertices $v_{|V(F)|+1},\dots,v_{p}$. For each pair of vertices $v_i,v_j, 1\le i<j \le p$ not contained in …
Let $Π_q$ be an arbitrary finite projective plane of order $q$. A subset $S$ of its points is called saturating if any point outside $S$ is collinear with a pair of points from $S$. Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to $\lceil\sqrt{3q\ln{q}}\rceil+ \lceil(\sqrt{q}+1)/2\rceil$. The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs.
An intersecting $r$-uniform straight line system is an intersecting linear system whose lines consist of $r$ points on straight line segment of $\mathbb{R}^2$ and any two lines share a point. Recently, the author [A. Vázquez-Ávila, \emph{On intersecting straight line systems}, J. Discret. Math. Sci. Cryptogr. Accepted] proved that any intersecting $r$-uniform straight line system $(P,\mathcal{L})$ has transversal number at most $ν_2-1$, with $r\geqν_2$, where $ν_2$ is the maximum cardinality of a subset of lines $R\subseteq\mathcal{L}$ such that every triplet of different elements of $R$ does …
We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\binom{n-2}{k-2}$ edges. This result can be viewed as a special case of the degree version of a well-known conjecture of Erdős on hypergraph matchings. Improving the work of Bollobás, Daykin, and Erd\H os from 1976, we show that given integers $n, k, s$ with $n\ge 3k^2 s$, every $k$-uniform hypergraph $H$ on $n$ vertices with minimum vertex degree greater than $\binom{n-1}{k-1}-\binom…
Let $H$ be an $n$-vertex 3-uniform hypergraph such that every pair of vertices is in at least $n/3+o(n)$ edges. We show that $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of $H$. The quantity two here is best possible and the degree condition is asymptotically best possible. This result also has an interpretation as the \emph{deficiency problems}, recently introduced by Nenadov, Sudakov and Wagner: every such $H$ can be made Hamiltonian by adding at most two vertices and all triples intersecting them.
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 Turán number ${\rm ex}_{\rm lin}(n,{\cal{F}})$ is the maximum number of edges in a $k$-uniform linear hypergraph $\mathcal H$ on $n$ vertices that does not contain any member of ${\cal{F}}$ as a subhypergraph. For $k\ge 2$ the $k$-fan $F^k$ is the $k$-uniform linear hypergraph having $k$ edges $f_1,\dots,f_k$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting all $f_i$ in a vertex different from $v$. We prove the follow…
