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Another easy criterion is the following: If ${\cal H} = (V, {\cal E})$ is a hypergraph such that $\bigcap {\cal E} \neq \emptyset$, then there can not be a graph edge set $E$ on $V$ such that ${\cal H} … are given any simple undirected graph $G=(V,E)$ on $V$, for every $v\in V$ the open neighborhood of $v$ does not contain $v$ - so the intersection of the edge set of the associated open neighborhood hypergraph …

An example of a hypergraph not representable by open nbhoods: Let $V$ be the set $\{1,2,3,4\}$ and ${\cal E}=\{V\setminus\{a,b\}: a<b\in V\}$ and set ${\cal H}=(V,{\cal E})$. …

(This is equivalent to saying that the hypergraph $(X,{\cal E})$ has chromatic number $2$.) …

Let $H=(V,E)$ be a hypergraph. If $\kappa$ is a cardinal, we say that a map $c:E\to \kappa$ is an edge coloring if whenever $e_1,e_2\in E$ with $e_1\cap e_2\neq \emptyset$ then $c(e_1)\neq c(e_2)$. … Given a positive integer $k$, is there a dense linear hypergraph $H= (V,E)$ with $V$ finite and $\chi_e(H) < 1/k\cdot |V|$? …

For any set $X$ and cardinal $\kappa$, we denote by $\ [X]^\kappa :=\ \binom X\kappa\ $ the collection of subsets of $X$ having cardinality $\kappa$. If $X$ is a set, we call a set system $E\subseteq …

Let $H = (V, E)$ be a hypergraph, that is $V$ is a set and $E \subseteq \mathcal{P}(V)$. …

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 …

A linear hypergraph is a hypergraph such that for all $e\neq e_1 \in E$ we have $|e\cap e_1|\leq 1$. For any positive integer $n$ let $[n] = \{1,\ldots,n\}$. … We say that $([n^2], E)$ is a square hypergraph if it is linear and every element of $E$ has $n$ elements. …

If $(X,\tau)$ is a topological space, we let the dense set hypergraph ${\cal D}(X,\tau)$ be the collection of all dense subsets of $X$ with respect to $\tau$. …

If $H=(V,E)$ is a hypergraph and $\kappa$ is a cardinal,we say a map $c:V\to\kappa$ is a coloring if the restriction $c\restriction_e$ of $c$ to $e$ is non-constant whenever $e\in E$ and $|e|>1$. …

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 \ …

Given a set $X$ and $k\in\mathbb{N}$ we call a subset of $X$ a $k$-subset if its cardinality is $k$. If ${\cal S}$ is a collection of subsets of $X$ and $x\in X$ we set ${\cal S}_x=\{S\in {\cal S}: x\ …

Let $H=(V,E)$ be a hypergraph such that for every $e\in E$ we have $|e|\geq 2$. … A map $c:V\to \kappa$, where $\kappa$ is a cardinal, is said to be a (hypergraph) coloring if for all $e\in E$ the restriction $c|_e$ is not constant. …

Let $n>1$ be an integer and let $[n]=\{1,\ldots,n\}$. An intersecting family on $[n]$ is a set $E\subseteq {\cal P}([n])$ such that for all $a,b\in E$ we have $a\cap b\neq\emptyset$. It is easy to see …

A hypergraph $H=(V,E)$ consists of a set $V$ and a collection of subsets $E \subseteq {\cal P}(V)$. … Is every hypergraph $H=(V,E)$ with $|V|\geq \omega$ and $|E| = |V|$ and $|e| = |V|$ for all $e\in E$ $2$-colorable? Motivation of question. …