Timeline for A question of Erdős
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 17, 2015 at 4:15 | answer | added | Gerry Myerson | timeline score: 3 | |
| Aug 2, 2015 at 20:07 | vote | accept | Ashutosh | ||
| Jul 25, 2015 at 1:38 | comment | added | Mirko | @MonroeEskew Thank you! I suspected that taking the union of lines (and not just any null-sets) might help avoid the use of MA, but didn't think of the details till after I read your message. | |
| Jul 25, 2015 at 0:17 | comment | added | Monroe Eskew | @Mirko, I think we can argue as follows. Suppose $A$ has positive measure and is the union of $< \frak c$ many lines. By Fubini's theorem there must be a positive subset $B$ of the $X$-axis such that for all $x \in B$, the vertical slice $A_x$ has positive linear measure. Choose such $x$ such that the vertical line $L_x$ at $x$ is not among the original lines. Then there is $y$ on $L_x \cap A$ that is not on any of the original lines, contradiction. | |
| Jul 24, 2015 at 22:58 | history | edited | Ashutosh | CC BY-SA 3.0 |
More questions no answers.
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| Jul 24, 2015 at 22:55 | comment | added | Mirko | @MonroeEskew I may need to verify the details but I am pretty sure a Bernstein set in the plane would work (hmm, with MA perhaps). Use that given any two points $A,B$, there are exactly two lines such that triangle $ABC$ has area $1$ iff $C$ is on one of these two lines. So given any perfect plane non-null set and any $<\mathfrak c$ (where $\mathfrak c=2^{\aleph_0}$) many points, we could remove all lines determined, as above, by pairs of these points ... we might need MA to conclude that the union of all these $<\mathfrak c$ many lines has measure $0$, to be able to carry on the construction. | |
| Jul 24, 2015 at 22:46 | comment | added | Ashutosh | Hi Monroe, it does, although your questions leads to some really interesting questions. I will edit to explain. | |
| Jul 24, 2015 at 22:24 | answer | added | Thomas Kalinowski | timeline score: 10 | |
| Jul 24, 2015 at 22:13 | comment | added | Monroe Eskew | Does it matter whether $A$ is measurable? Is there a counterexample of $A$ with outer measure $\infty$ and inner measure 0? | |
| Jul 24, 2015 at 20:52 | history | edited | GH from MO | CC BY-SA 3.0 |
edited body; edited title
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| Jul 24, 2015 at 20:35 | history | edited | Ashutosh | CC BY-SA 3.0 |
added 12 characters in body
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| Jul 24, 2015 at 20:29 | history | asked | Ashutosh | CC BY-SA 3.0 |