Timeline for From very many sets of fixed measure in a probability space, can we select many that have a positive intersection?
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12 events
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| Sep 30, 2012 at 10:09 | comment | added | Jakob | Douglas: You are right that Ramsey's Theorem gives ridiculously bad bounds. The trivial proof that Bill Johnson pointed out for the Lemma in the original question seems to give something like $M=2\ell/\delta$ and $\varepsilon = \delta/(2 M)^\ell$. So we get a "good" lower bound for $M$, but if we choose a very large $M$ then $\varepsilon$ will be "bad". The Ramsey proof for the Lemma in my "answer" below gives a $\varepsilon$ independent of $M$, e.g. $\varepsilon(\delta,k)=\delta/(10)^k$, and something ridiculously large for $M$ (which we get out of Ramsey's Theorem). | |
| Sep 29, 2012 at 21:23 | answer | added | Jakob | timeline score: 3 | |
| Sep 29, 2012 at 21:13 | comment | added | Douglas Zare | What bounds are you supposed to get using Ramsey theory? The ones I get from Ramsey theory seem really bad compared to what I expect is true. | |
| Sep 29, 2012 at 8:20 | comment | added | Ed Dean | Jakob, you might find the closely related Theorem 2.2 of the following paper to be of interest: dx.doi.org/10.4115/jla.2012.4.3 | |
| Sep 29, 2012 at 7:12 | answer | added | Robin Tucker-Drob | timeline score: 1 | |
| Sep 29, 2012 at 0:13 | comment | added | Bill Johnson | Sure, but then some collection of $\ell$ sets contains many points (not that this gives a very good estimate). | |
| Sep 28, 2012 at 23:28 | comment | added | Goldstern | @bill johnson: I think this only gives you a large set of points that are in at least $\ell$ sets, but Jakob wants that they are all in the same $\ell$ sets. | |
| Sep 28, 2012 at 23:22 | comment | added | Bill Johnson |
Analyse the sum of the characteristic functions of the sets, which has integral at least $|M| \delta$ and sup norm at most |M|$.
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| Sep 28, 2012 at 22:28 | comment | added | Andrés E. Caicedo | Welcome, Jakob. | |
| Sep 28, 2012 at 22:16 | history | edited | Goldstern |
reference-request
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| Sep 28, 2012 at 22:13 | comment | added | Goldstern | Welcome to mathoverflow! What a coincidence, I was just thinking about this very question :-) | |
| Sep 28, 2012 at 21:53 | history | asked | Jakob | CC BY-SA 3.0 |