Timeline for Cycling through a general combinatorial design on $\omega$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 28, 2022 at 8:05 | vote | accept | Dominic van der Zypen | ||
| Oct 28, 2022 at 8:04 | history | edited | Dominic van der Zypen | CC BY-SA 4.0 |
added 56 characters in body
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| Oct 28, 2022 at 8:04 | comment | added | Dominic van der Zypen | Right - I want infinitely many blocks, will add this to the question description. Now I am looking forward to reading Joel's answer! | |
| Oct 28, 2022 at 7:37 | answer | added | Joel David Hamkins | timeline score: 4 | |
| Oct 28, 2022 at 3:03 | comment | added | bof | I think "infinitely many blocks" is a necessary and sufficient condition. For sufficiency, it looks offhand like the straightforward inductive construction works. | |
| Oct 28, 2022 at 2:53 | comment | added | bof | @Holo More generally, if such a sequence of bijections exists, the number of blocks must be infinite. This is because, for any given $k\in\omega$, for sufficiently large $n$ the numbers $\varphi_n(1),\varphi_n(2),\dots,\varphi_n(k)$ will all be in different blocks. | |
| Oct 27, 2022 at 21:47 | comment | added | Holo | If there exists a cofinite block there is no such sequence of bijections | |
| Oct 27, 2022 at 19:18 | comment | added | Joel David Hamkins | What I mean to say is that if there is only one block, then one will fail the uniqueness claim on $n$ in the desired property, since every $\varphi_n$ will work in this case. | |
| Oct 27, 2022 at 19:11 | comment | added | Joel David Hamkins | What if there is just one big block? Perhaps you want to assume there are infinitely many blocks? | |
| Oct 27, 2022 at 19:00 | history | asked | Dominic van der Zypen | CC BY-SA 4.0 |