Timeline for Refinement-minimal intersecting covers
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 23 at 11:18 | vote | accept | Dominic van der Zypen | ||
| Apr 23 at 8:35 | answer | added | Taras Banakh | timeline score: 5 | |
| Apr 22 at 10:50 | vote | accept | Dominic van der Zypen | ||
| Apr 22 at 10:51 | |||||
| Apr 22 at 10:13 | answer | added | bof | timeline score: 2 | |
| Apr 19 at 7:44 | comment | added | Dominic van der Zypen | Ah, yes, thanks @bof, that's what I meant, but I wrote it in a wrong way. | |
| Apr 19 at 2:58 | comment | added | bof | If $\mathcal C$ is an intersecting cover of $X$ which is linear and has the property that, for each $A\in\mathcal C$, there is a unique element $a\in A$ which is covered by no other element of $\mathcal C$, then $\mathcal C$ is a minimal cover. Is that what you meant? | |
| Apr 19 at 0:53 | comment | added | bof | As an example of an intersecting cover of $\omega$ by finite sets which has no linear refinement, let $\mathcal C$ consist of the sets $\{0,2,4\}$, $\{1,3,4\}$, and $\{0,1,n\}$ where $n\ge5$. Note that $\mathcal C$ is a refinement-minimal cover and is not linear. | |
| Apr 18 at 20:40 | comment | added | Dominic van der Zypen | @bof first I was convinced that for the case you mention, even a stronger statement holds. Let us call an intersecting cover ${\cal C}$ linear if $|A\cap B| = 1$ for all $A\neq B\in{\cal C}$. I thought that in the case you mention, every such cover has a linear refinement (which would be refinement-minimal -> but now I am not convinced). | |
| Apr 18 at 3:14 | comment | added | bof | Is there an obvious answer in the special case where $X=\omega$ and $\mathcal C$ is an intersecting cover whose elements are finite sets? In this case I don't think Zorn's lemma or the axiom of choice would come up. But maybe this case is trivial and I'm just not clever enough to see it. | |
| Apr 17 at 8:48 | answer | added | HenrikRüping | timeline score: 3 | |
| Apr 17 at 8:18 | comment | added | HenrikRüping | I would try to approach this with Zorns lemma. I would like to consider the set of intersecting covers of $X$ as a poset, but as you stated, refining is not antisymmetric. Thus we can pass to equivalence classes of intersecting covers where $C\sim D$ iff $C\le D$ and $D\le C$. Then I would think about the assumption on chains in Zorns lemma. | |
| Apr 17 at 7:14 | history | edited | Dominic van der Zypen | CC BY-SA 4.0 |
reformulated question in a clearer way
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| Apr 16 at 16:40 | history | asked | Dominic van der Zypen | CC BY-SA 4.0 |