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Hypergraphs are generalizations of graphs, where edges can be made of more than two vertices.

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Minimal covers in hypergraphs with finite edges

Let $H=(V,E)$ be a hypergraph. We say that $C\subseteq E$ is a cover if $\bigcup C = V$. Let $H$ be a hypergraph with the following properties: $\bigcup E = V$, all members of $E$ are finite, and $d,...
2
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1answer
59 views

Example of self-dual hypergraph with infinite edges

What is an example of a hypergraph $H=(V,E)$ with $|e|\geq \aleph_0$ for all $e\in E$ and the property that $H\cong H^*$ where $H^*$ is the dual hypergraph of $H$?
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1answer
140 views

A set-family game

Two players, Green and Red, play a zero-sum game. It is parametrized by two integers $n\geq 0, k\geq 0$, and a finite family $F$ of sets of size $n$ (each set may appear multiple times in $F$). Each ...
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49 views

Clique number for hypergraphs

Does anybody know any link/source where I can find examples of hypergraphs with their clique numbers? I need a few examples to test an algorithm and do not want to go for randomly generated hypergraph....
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2answers
142 views

Maximum intersecting set families of $\{1,\ldots,n\}$

Let $n\geq 3$ be an integer. We call a family ${\cal C}$ of subsets of $\{1,\ldots,n\}$ intersecting if it has the following properties: $A, B\in {\cal C}$ implies $A\cap B \neq \emptyset$, and $A \...
8
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1answer
188 views

A generalization of Erdős-Ko-Rado theorem

Is there any result known about the following generalization of the Erdős-Ko-Rado theorem? Let $n, k, r, s$ be positive integers. We call a family $\mathcal{F}$ of $k$-element subsets of $\{1,\ldots, ...
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1answer
51 views

Strong and weak chromatic number of infinite bounded hypergraphs

This is a follow-up of an older question. Let $H = (V, E)$ be a hypergraph such that every member of $E$ has more than $1$ element. Let $\kappa$ be a cardinal. We say that $c: V\to \kappa$ is a weak ...
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0answers
80 views

Does anybody know how to prove or disprove the following guess about edge coloring of Hypergraphs?

Definition. Suppose $H=(V,E)$ is a hypergraph. Call a hyperedge $e=\{v_1,v_2,\dotsc,v_k\}$ a $k$-star if in the 1-skeleton of $H$ there is a copy of $K^{1,k-1}$ whose union equals $e$. (Obviously in ...
1
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1answer
58 views

Strong and weak chromatic number of infinite hypergraphs of finite rank

Let $H = (V, E)$ be a hypergraph such that every member of $E$ has more than $1$ element. Let $\kappa$ be a cardinal. We say that $c: V\to \kappa$ is a weak coloring if for all $e \in E$ the ...
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1answer
80 views

Why not have many edges instead of one edge connecting to many nodes (in hypergraphs) [closed]

As a follow-up to What are the Applications of Hypergraphs, the linked article: Learning with Hypergraphs: Clustering, Classification, and Embedding Mentions this: Let us consider a problem of ...
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0answers
93 views

Set version of ramsey type problem

For two sets of numbers $A,B$, write $A<B$ iff $\max A<\min B$. For a sequence of integers $a_0,\cdots,a_{n-1}>0$, let $Prop(a_0,\cdots,a_{n-1})$ denote the following proposition: Given $n$ ...
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1answer
299 views

Ramsey type theorem

Let $\mathcal{P}(\{0,\dotsc,7\})$ denote the power set of $\{0,\dotsc,7\}$. Is the following true? For any function $f: \mathcal{P}(\{0,\dotsc,7\})\rightarrow\{0,1\}$ there exists $0\leq k\leq 3$ ...
4
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1answer
130 views

Complexity for calculating number of Perfect Matchings in k-regular hypergraph

Let $G(V,E)$ be a unweighted, k-regular hypergraph, with vertices $V=(v_1, ... v_n)$ and edges $E=(e_1, ... e_m)$. The k-regularity leads to $|e_i|=k$ (i.e. every edge contains exactly $k$ vertices). ...
7
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2answers
309 views

Infinite projective plane with small edges

Let $\kappa$ be an infinite cardinal. We say $E\subseteq {\cal P}(\kappa)$ is an infinite projective plane on $\kappa$ if $e_1\neq e_2\in E$ implies $|e_1\cap e_2| = 1$, and whenever $n\neq m\in \...
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1answer
80 views

Partitioning finite hypergraphs with edges [closed]

Let $H=(V,E)$ be a hypergraph such that $|V|$ is infinite, and the following statements hold: if $a\neq b\in E$ then $|a\cap b|\leq 1$, and every vertex $v\in V$ is contained in at least $2$ members ...
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1answer
111 views

A weak version of the Erdös-Faber-Lovasz conjecture

A hypergraph is a pair $H=(V,E)$ where $V$ is a nonempty set, and $E\subseteq {\cal P}(V)\setminus\{\emptyset\}$ is a collection of non-empty subsets of $V$. Strong colorings. If $\kappa$ is a ...
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0answers
44 views

An equation involving fractional covering number of hypergraphs

Let $\mathcal{H}=(S,\mathcal{X})$ be a hypergraph, where $S = \{ s_1, \ldots, s_n \}$, and $\mathcal{X} = \{ X_1, \ldots, X_m \}$. The dual hypergraph $\mathcal{H}^*$ of $\mathcal{H}$ is the ...
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1answer
131 views

Choice sets in “thick” sets of sets

Let $X\neq \emptyset$ be a set and let ${\cal S} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subsets of $X$. We say $C\subseteq X$ is a choice set for ${\cal S}$ if for ...
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1answer
57 views

“Popular” vertices in an infinite hypergraph

Let $\kappa$ be an infinite ordinal, and suppose $E\subseteq {\cal P}(\kappa)$ with $|E| > \kappa$. Let us call $x\in \kappa$ popular if $$|\{e\in E: x\in e\}| > \kappa.$$ If $\text{Pop}(\kappa)...
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2answers
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Known result about existence of $n$-vertex $k$-uniform $r$-hypergraphs?

Are there known results about $n,k,r$ such that $n$-vertex $k$-uniform $r$-regular hypergraphs exist? If this is too large a class of hypergraphs, what if $k=\tilde{\theta}(\sqrt{n})$? What if an ...
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2answers
124 views

Small dense subsets in “Hausdorff” hypergraphs

Let $H=(V,E)$ be a hypergraph. We call it Hausdorff if for all $x\neq y \in V$ there are $e_1,e_2\in E$ with $e_1\cap e_2 = \emptyset$ such that $x\in e_1$ and $y\in e_2$. We say that $D\subseteq V$ ...
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2answers
106 views

Infinite “$T_1$”- hypergraphs

Let $X$ be an infinite set, and let $E \subseteq {\cal P}(X)$ be a collection of subsets of $X$. We say that $E$ is $T_1$ (with respect to $X$) if for all $x\neq y\in X$ there is $e\in E$ with $x\in e$...
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2answers
123 views

Coloring hypergraphs with no singleton intersections

Let $H=(V,E)$ be a hypergraph. If $\kappa>0$ is a cardinal, we say the hypergraph $H$ is $\kappa$-chromatic if there is a function $c:V\to\kappa$ such that for all $e\in E$ the restriction $c|_e$ ...
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0answers
107 views

Infinite $3$-chromatic hypergraphs

Let $H=(V,E)$ be a hypergraph. If $\kappa>0$ is a cardinal, we say the hypergraph $H$ is $\kappa$-chromatic if there is a function $c:V\to\kappa$ such that for all $e\in E$ the restriction $c|_e$ ...
2
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1answer
78 views

How to encode subgraphs as hyperedges

Hi i was reading a paper "Propagating Distributions on a Hypergraph by Dual Information Regularization by Koji Tsuda", and one section stood out to me. hypergraphs have more flexibility in ...
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1answer
47 views

Inequality about the minimum vertex degree in $k$-uniform hypergraphs

Let $H=(V,E)$ be a $k$-uniform hypergraph with $n$ vertices, that is, $V:=V(H)$ is a $n$-element finite set of vertices and $E:=E(H)\subset\binom{V}{k}$ is a family of $k$-element subsets of $V$. ...
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104 views

Edge-coloring number of a linear hypergraph

A linear hypergraph is a hypergraph $H=(V,E)$ such that $|e|\geq 2$ for all $e\in E$, $|e_1\cap e_2|\leq 1$ for all $e_1, e_2\in E$ with $e_1\neq e_2$. An edge coloring of $H$ is a function $c:E\to ...
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3answers
241 views

Non-isomorphic hypergraphs on $\omega$

Let $H_i = (V_i, E_i)$ be hypergraphs for $i=1,2$. Then we say that $H_1\cong H_2$ if there is a bijection $\varphi:V_1\to V_2$ such that $A\in E_1$ implies $\varphi(A) \in E_2$ and $B\in E_2$ implies ...
6
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1answer
237 views

Is an $O(n^{d-1})$ bound known for the maximum number of edges in an ordered $n$-vertex hypergraph avoiding a fixed $d$-permutation hypergraph?

By a $d$-permutation hypergraph, I mean, for some fixed integer $k$, a $d$-uniform hypergraph on $[dk]$ with $k$ disjoint edges such that every edge has exactly one vertex from each of $\{1,\ldots,k\}$...
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Non-projective linear hypergraphs

A linear hypergraph is a pair $\pi=(\{1,\ldots,n\}, L)$ where $n\in\mathbb{N}$, $n\geq 2$ and $L\subseteq {\cal P}(X)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \...
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0answers
81 views

Hypergraphs with edge coloring number equalling number of vertices

A linear hypergraph is a pair $\pi=(\{1,\ldots,n\}, L)$ where $n\in\mathbb{N}$, $n\geq 2$ and $L\subseteq {\cal P}(X)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \...
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0answers
86 views

Steiner-like systems with large edges and many intersections

Let $l\geq 3$ be an integer. Is there $n\in\mathbb{N}$ and a hypergraph $H=(\{1,\ldots,n\},E)$ with the following properties? for all $e\in E$ we have $|e| \geq l$ $e_1\neq e_2 \in E \implies |e_1 \...
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1answer
513 views

3-Approximation Algorithm for 3-Hitting Set

I need to find a $3$-approximation algorithm for finding a $3$-hitting set. The set-up is that I have a set $S$ and a family $\mathcal{F}$ of subsets of $S$, where each member of $\mathcal{F}$ ...
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0answers
152 views

Which weighted directed hypergraphs have incidence matrix of full rank?

what are the necessary conditions for a weighted directed hypergraph to have an incidence matrix of full rank? In this context, we can define the incidence matrix as follows: Let $V = \{v_1,v_2,...,...
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0answers
80 views

Shortest hyperpath algorithm in intuitionistic fuzzy hypergraphs

I was looking for an algorithm to calculate the shortest hyperpath in intuitionistic fuzzy hypergraphs and I found only this article (which propose two algorithms). Are there any others algorithms ...
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1answer
49 views

Cardinalities of saturated linear hypergraphs

A saturated linear hypergraph is a hypergraph $H=(V,E)$ such that $|e|\geq 2$ for all $e\in E$, $|e_1\cap e_2| = 1$ for all $e_1, e_2\in E$ with $e_1\neq e_2$, and $|\{e\in E:v\in e\}| = 2.$ Let $E$...
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1answer
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Complete and saturated linear hypergraphs

A linear hypergraph is a hypergraph $H=(V,E)$ such that $|e|\geq 2$ for all $e\in E$, $|e_1\cap e_2|\leq 1$ for all $e_1, e_2\in E$ with $e_1\neq e_2$. We call a linear hypergraph complete if there ...
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3answers
140 views

Minimal number of edges for complete linear hypergraphs

A complete linear hypergraph is a hypergraph $H=(V,E)$ such that $|e|\geq 2$ for all $e\in E$, $|e_1\cap e_2|=1$ for all $e_1, e_2\in E$ with $e_1\neq e_2$, and for all $v\in V$ we have $|\{e\in E:v\...
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1answer
53 views

Maximizing the chromatic number of regular hypergraphs

Let $H=(V,E)$ be a hypergraph, let $n\in\mathbb{N}$. A vertex coloring is a map $c: V\to \{1,\ldots, n\}$ such that for $v\neq w \in V$ we have $c(v)\neq c(w)$ whenever there is $e\in E$ such that $v, ...
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1answer
587 views

Edge chromatic number of hypergraphs

This is question Selection problem in a collection of non-empty sets with a simplification in criterion 3. Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal P}(X)\setminus\{\...
2
votes
1answer
232 views

Proof that it's possible to colour all elements in set, that all subsets will be bicolored

(For my easy understanding, let me rewrite the question. The author should feel free to remove my edit or... accept it; I am leaving the original formulation at the end intact). ================= ...
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3answers
230 views

Class of hypergraphs that are always the neighborhood hypergraph of some simple graph

Let $G=(V,E)$ be a graph. Its (open) neighborhood hypergraph $\mathcal{H}(G)$ has the same vertex set $V$ with a hyperedge for the (open) neighborhood of every vertex $v \in V$. It seems that not ...
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1answer
222 views

Algorithm for 2-coloring classes of 3-uniform hypergraphs

Hi all and thanks in advance for your efforts. I'm interested in 2-coloring 3-uniform hypergraphs. I know that in general, the problem of deciding if a 3-uniform hypergraph is 2-colorable is NP-hard....
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1answer
165 views

Is it true that any $3$-uniform hypergraph that is not $k$-colorable must have $\Omega(k^3)$ edges?

What is the best lower bound in terms of $k$ on the number of edges in a $3$-uniform hypergraph that is not $k$-colorable? Thanks in advance.
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1answer
34 views

Maximal expansions of strongly minimal covers of hypergraphs

Let $H = (V,E)$ be a hypergraph, that is $V$ is a set and $E \subseteq {\cal P}(V)$. We assume $\bigcup E = V$. Moreover we assume that every $e\in E$ is contained in some maximal member $e'\in E$ (...
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votes
1answer
150 views

Is there a Degenerate Dependency Local Lemma?

The Lovasz Local Lemma has several generalizations, with names usually starting with L, such as Lopsided or Lefthanded. Here I ask whether another possible generalization (for which I could not yet ...
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0answers
82 views

Is the size of maximum matching in vertex transitive 3-uniform hyper-graph on $n$ vertices always $\Omega(n)$?

What is the best known lower bound on the size of the maximum matching in a vertex transitive $3$-uniform hyper-graph?
3
votes
3answers
132 views

Linear intersection number and vertex covering number

A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \in L$ then $|...
1
vote
1answer
110 views

Minimality condition in a certain class of hypergraphs

A hypergraph is a pair $H=(V,E)$ such that $V$ is a (possibly infinite) set and $E\subseteq \mathcal{P}(V)$. $C\subseteq E$ is said to be a cover if $\bigcup C = V$ and $C$is minimal if $C'\subseteq C$...
4
votes
1answer
158 views

Strongly minimal covers

Let $H=(V,E)$ be a hypergraph, that is $V$ is a set and $E\subseteq \mathcal{P}(V)$. We say that $C\subseteq E$ is a cover of $H$ if $\bigcup C = V$. A cover $M\subseteq E$ is said to be strongly ...