Timeline for Partition regular systems: do they have solution in (very dense) set of integers?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 15, 2015 at 6:23 | comment | added | Sean Eberhard | Maybe a good way of formulating such a result is like this: Suppose there are $d$ linearly independent positive solutions. Then there is some $\epsilon>0$ depending on $A$ and $d$ such that as $n$ becomes large compared to everything else then whenever $X\subset\{1,\dots,n\}$ has density at least $1-\epsilon$ then there are at least $\epsilon n^d$ solutions in $X$. My guess is this is true by the same proof. | |
| Dec 14, 2015 at 10:17 | comment | added | Johnny Cage | Thanks for the answer! I was wondering if this result can be strengthened to obtain a robust version of the existence of solutions, as it happens with Varnavides theorem when dealing with arithmetic progressions. | |
| Dec 2, 2015 at 14:47 | vote | accept | Johnny Cage | ||
| Dec 2, 2015 at 14:46 | comment | added | Sean Eberhard | Maybe I'd say it comes up more often in measure-theoretic arguments, since outer regularity implies that an arbitrary set of positive measure has arbitrarily large density in some interval. Therefore for example every set of positive measure contains an affine translate of every finite set. | |
| Dec 2, 2015 at 14:42 | comment | added | Sean Eberhard | Probably. I think I remembered it from the proof that a set without, say, $10$-term geometric progressions must have density strictly less than $1$. | |
| Dec 2, 2015 at 14:36 | comment | added | Ben Barber | Sean, does this argument come up often in additive combinatorics? I've used it before (to dodge sparse colour classes when proving Ramsey results) but haven't until now seen it elsewhere. | |
| Dec 2, 2015 at 14:22 | history | answered | Sean Eberhard | CC BY-SA 3.0 |