Timeline for Sunflowers in maximal almost disjoint families
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Jun 10, 2021 at 11:27 | comment | added | Joel David Hamkins | Regarding KP Hart's comment, I was using $\frak{a}_\kappa$ differently, to refer to almost disjoint families on $\omega$, but omitting any size $\kappa$ sunflower. | |
| Jun 10, 2021 at 8:12 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 10, 2021 at 8:10 | comment | added | Joel David Hamkins | I have now asked the question at: mathoverflow.net/q/394987/1946 | |
| Jun 10, 2021 at 6:19 | comment | added | KP Hart | I take it $\mathfrak{a}_\kappa$ refers to the adness number of $\kappa$ modulo $[\kappa]^{<\kappa}$? You'd expect it to increase with $\kappa$ but $\mathfrak{a}_{\aleph_1}<\mathfrak{c}$ is consistent. Baumgartner: Independence proofs and combinatorics | |
| Jun 9, 2021 at 21:28 | comment | added | Dominic van der Zypen | Thanks Joel for your addition on the $\alpha_\kappa$'s -- and I definitely think your additional question deserves its own post! | |
| Jun 9, 2021 at 19:04 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 9, 2021 at 18:18 | comment | added | Joel David Hamkins | I got confused about $\frak{a}_\kappa$ and $\frak{a}_\lambda$ when $\kappa<\lambda$. At first I thought there should be an obvious relation there, but now it seems there is no trivial argument there. Can someone straighten me out on that? | |
| Jun 9, 2021 at 18:17 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 9, 2021 at 18:11 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 9, 2021 at 18:07 | comment | added | Joel David Hamkins | Perhaps I shall ask my question at the end as an MO question. Vote up this comment if you would want that to happen. | |
| Jun 9, 2021 at 18:04 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 9, 2021 at 8:21 | comment | added | KP Hart | You say Freiling, I say Kuratowski: eudml.org/doc/213244 | |
| Jun 9, 2021 at 8:14 | comment | added | Joel David Hamkins | Perhaps one can hope to get larger families by moving from $n=3$ to $n=4$ and higher? It would be sort of like the situation with Freiling's axiom. | |
| Jun 9, 2021 at 7:49 | comment | added | KP Hart | I just read a paper that has an argument in it that shows it stops at $\aleph_1$. I'll write a second answer later. Oh no, I misspoke. | |
| Jun 9, 2021 at 7:47 | comment | added | Joel David Hamkins | The particular feature I used--a mad family where every set has distinct intersections with all earlier sets, is impossible for families bigger than $\aleph_1$. | |
| Jun 9, 2021 at 7:45 | comment | added | Joel David Hamkins | It seems that plenty is still open here. Can we do it in ZFC? Can we do it with larger mad families? | |
| Jun 9, 2021 at 7:43 | vote | accept | Dominic van der Zypen | ||
| Jun 9, 2021 at 8:40 | |||||
| Jun 9, 2021 at 7:41 | comment | added | KP Hart | Yes, and opens up the question about m.a.d. families of size larger than $\aleph_1$. | |
| Jun 9, 2021 at 7:36 | comment | added | Joel David Hamkins | Ah, that would make it consistent with the failure of CH? | |
| Jun 9, 2021 at 7:35 | comment | added | KP Hart | You can combine this with the proof of Theorem VIII.2.3 in Kunen's book to make it Cohen indestructible. | |
| Jun 9, 2021 at 7:17 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |