Timeline for Rank of sumsets in matroids
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 6, 2022 at 2:53 | vote | accept | Shahab | ||
| Sep 12, 2021 at 22:52 | vote | accept | Shahab | ||
| Sep 13, 2021 at 1:25 | |||||
| Sep 12, 2021 at 22:51 | vote | accept | Shahab | ||
| Sep 12, 2021 at 22:51 | |||||
| Sep 12, 2021 at 22:51 | history | bounty ended | Shahab | ||
| Sep 10, 2021 at 20:29 | comment | added | Shahab | Take a look at Theorem 2.1 of core.ac.uk/download/pdf/301643726.pdf , a matroid analog of Kneser is given but it gives something weaker than what I've mentioned. | |
| Sep 8, 2021 at 17:22 | comment | added | Shahab | Yes @Antoine Labelle, you are right. I however hope that for certain subsets of probably a very small family of matroids it might be the case. Indeed, for situations that $X$ and $Y$ are not full-rank but $X+Y$ is close to being full-rank? | |
| Sep 7, 2021 at 11:37 | history | edited | Antoine Labelle | CC BY-SA 4.0 |
added 371 characters in body
|
| Sep 6, 2021 at 19:13 | comment | added | Shahab | Another counterexample: Take a sum-free subset of $A$ of $G$, and let $M$ be a matroid whose vertices set $V(M)$ is $A$. Set $X=Y=A$. Let $r(M)=n$. Then we have: $r(X+Y)=0$ but $r(x)=r(Y)=n$, and $r(H)\leq n$. This violates the claimed inequality | |
| Sep 4, 2021 at 22:28 | comment | added | Shahab | The matroid is defined over a group, so sumsets are all belong to the ground set, $G$. The matroid itself may not be closed under taking pairwise sumsets. That is, if $X+Y$ is not an independent set, it is legit to discuss $r(X+Y)$ (which is equal to the cardinality of the largest subset of $X+Y$ that is independent in $M$) @Jukka Kohonen | |
| Sep 4, 2021 at 21:10 | comment | added | Jukka Kohonen | Is it understood, in the first place, that the matroid is closed under taking pairwise sumsets? And that its structure is compatible with the subset relation of the sets in $G$? | |
| Sep 4, 2021 at 16:55 | comment | added | Shahab | Thanks, Antonie. I just edited my question. I agree that assuming nothing on the group structure and no restriction on the matroid will make things very wild. | |
| Sep 4, 2021 at 14:51 | history | answered | Antoine Labelle | CC BY-SA 4.0 |