Search Results

Let $H=(V,E)$ be a hypergraph. We say that $H$ is thin if for every $v\in V$ the set $E_v=\{e\in E:v\in e\}$ is finite. … Does every thin hypergraph $H=(V,E)$ with $\bigcup E = V$ have a minimal dominating set $D_0\subseteq V$? …

Every hypergraph $(\omega, E)$ where $E$ is countable and consists of infinite sets has property $\newcommand{\B}{\mathbf{B}}\B$. … A hypergraph $H=(V,E)$ is said to have property $\B$ if there is $B\subseteq V$ such that $$e\cap B\neq\emptyset\neq e\setminus B$$ for all $e\in E$ with $|e|>1$. …

Does every hypergraph that does not admit a complete minor with $5$ elements have a coloring with $4$ colors? Below are the definitions to make this precise. … If $H = (V, E)$ is a hypergraph and $W \subseteq V$, then we let the induced sub-hypergraph of $W$ be $H|_W := (W, E|_W)$, where $E|_W := \{e \cap W: e \in E \text{ and }e\cap W \neq \emptyset\}$. …

Let $H = (V,E)$ be a hypergraph. For $v\in V$ we set $E_v = \{e\in E: v\in E\}$. The dual of $H$ is defined by $H^* =(E, V^*)$ is, where $V^* = \{E_v:v\in V\}$. …

Let $H = (V,E)$ be a hypergraph. For $v\in V$ we set $E_v = \{e\in E: v\in E\}$. The dual of $H$ is defined by $H^* =(E, V^*)$ is, where $V^* = \{E_v:v\in V\}$. … If $\kappa,\lambda >1$ are cardinals, is there necessarily a hypergraph $H$ with $\chi(H) = \kappa$ and $\chi(H^*) = \lambda$? …

, for every connected hypergraph, we have $\bigcup E = V$.) … If $H =(V,E)$ is a linear connected hypergraph, is there $E_0\subseteq E$ with the following properties? …

For any set $X$, let $[X]^2 = \big\{\{a,b\}:a\neq b\in X\big\}$. If $\kappa>1$ is a cardinal, then a splitting family is a collection ${\cal S} \subseteq {\cal P}(\kappa)$ such that for every $Q \in [ …

Motivation. Yesterday I was sitting idly in the train, contemplating the train network. I noticed that a lot of lines (not all) intersected, and some pairs of lines intersected in quite a few stations …

If $H=(V,E)$ is a hypergraph then the chromatic number $\chi(H)$ is defined to be the least cardinal $\kappa \leq |V|$ such that there is a map $c:V\to \kappa$ such that for all $e\in E$ with $|e| \geq …

This question is loosely inspired by the exact cover / partition problem in computer science. Let $X\neq \emptyset$ be a set and let ${\cal E}\subseteq {\cal P}(X)$. For $x\in X$ we let $c_{\cal E}(x) …

Is there a set $E\subseteq {\cal P}(\mathbb{N})$ of subsets of $\mathbb{N}$ with the following properties? $|e| > 2$ for all $e\in E$, $e_1\neq e_2 \in E$ implies $|e_1 \cap e_2| \leq 1$, for all $m, …

$\newcommand{Po}{{\cal P}(\omega)}$ $\newcommand{lh}{\leq_{\text{hom}}}$ If $H_i = (V_i, E_i)$ are hypergraphs for $i = 1,2$, then a map $f:V_1 \to V_2$ is said to be a (hypergraph) homomorphism if $f( …

This is a kind of "dual" of an older question. Is there a finite family ${\frak F}\subseteq {\cal P}(\mathbb{N})$ such that for all $a\neq b\in\mathbb{N}$ there is $S\in{\frak F}$ with $|S\cap \{a,b\} …

A hypergraph $H= (V,E)$ is said to be bipartite if there is $S\subseteq V$ such that for all $e\in E$ with $|e|>1$ we have $$S\cap e\neq \emptyset\neq (e\setminus S).$$ A hypergraph is said to be linear … if distinct edges intersect in at most $1$ element, and we say the hypergraph is $\aleph_0$-uniform if every edge contains countably infinite elements. …

Motivation. An intersecting family is a collection of subsets ${\cal S}\subseteq {\cal P}(X)$ of a set $X\neq \emptyset$ such that $A\cap B\neq \emptyset$ for all $A,B\in {\cal S}$. The intersections …