All Questions
Tagged with covering set-theory
13 questions
10
votes
1
answer
262
views
Does every linear cover contain a minimal cover?
This is a follow-up question to an older question.
Let $X\neq \emptyset$ be a set. We say that ${\cal C}\subseteq {\cal P}(X)$ is a cover if $\bigcup {\cal C} = X$, and we call ${\cal C}$ linear if $|...
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5
votes
1
answer
158
views
(Weakly) minimal subcovers of linear covers
Motivation. The starting point of this question is the trivial observation that if we cover $\mathbb{N}$ with $$\big\{\{0,\ldots n\}: n\in \mathbb{N}\big\},$$ then this cover doesn't have a minimal ...
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6
votes
3
answers
236
views
Refinement-minimal intersecting covers
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....
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1
vote
1
answer
92
views
Hypergraphs with finite matching / covering balance
Let $H=(V,E)$ be a hypergraph such that $\emptyset\notin E$. We say that $C\subseteq V$ is a (vertex) cover if for all $e \in E$ we have $C\cap e\neq \emptyset$. The minimum size that a cover can have ...
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4
votes
1
answer
193
views
Is König's Property for graphs inheritable from finite subgraphs?
Let $G = (V,E)$ be a simple, undirected graph. A set $C \subseteq V$ is said to be a (vertex) cover if $C \cap e \neq \emptyset$ for all $e\in E$. A matching is a set $M\subseteq E$ of pairwise ...
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0
votes
0
answers
81
views
Efficiency of covers
Let $X\neq \emptyset$ be a set. We say $C \subseteq {\cal P}(X)$ is a cover of $X$ if $\bigcup C = X$. For covers $C, D$ of $X$ we say that $C$ is more efficient than $D$ if $|C\setminus D| < |D \...
- 48.9k
3
votes
2
answers
125
views
Avoiding multiply covered vertices in graph edge coverings
Let $G=(V,E)$ be a simple, undirected graph with $\bigcup = E$ (that is, there are no isolated vertices). We say that $C\subseteq E$ is an edge cover of $G$ if $\bigcup C = V$. For any edge cover $C$ ...
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1
vote
1
answer
154
views
On a combinatorial set covering property
Let $\kappa < \lambda < \mu$ be infinite cardinals. Is there a collection ${\cal U}\subseteq {\cal P}(\mu)$ of subsets of $\mu$ with the following properties?
for all $U\in {\cal U}$ we have $|...
- 48.9k
1
vote
1
answer
83
views
Choice sets in covers with small intersections
Let $X\neq \emptyset$ be a set. We say ${\cal C} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ is a cover of $X$ if $\bigcup {\cal C} = X$. A subset $S\subseteq X$ is a choice set for ${\cal C}$ if $|S\...
- 48.9k
2
votes
1
answer
109
views
Edge covers in infinite graphs
If $G=(V,E)$ is a simple, undirected graph, then $C\subseteq V$ is an edge cover if $C\cap e \neq \emptyset$ for all $e\in E$.
The "best" covers in some sense are subsets $C\subseteq V$ that meet ...
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7
votes
3
answers
492
views
Minimal covering sets in families of sets intersecting in at most $1$ point
Let $X$ be an infinite set, and let ${\cal A}\subseteq{\cal P}(X)$ be a family of non-empty sets. We say $S\subseteq X$ is a cover for ${\cal A}$ if $A\cap S \neq \emptyset$ for all $A\in{\cal A}$.
...
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2
votes
0
answers
149
views
Cardinality of compact doubling metric spaces with fast growing covering numbers
In this question it was established that if the growth of the number of branches of an at-most $k$-branching tree is $\Omega(k^n)$ (in the Knuth sense), then the tree has continuum many branches.
...
- 13.2k
4
votes
1
answer
245
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 ...
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