Timeline for minimum number of bases of a matroid, that comes from a convex polytope
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 14, 2016 at 22:59 | vote | accept | Mr Shorly | ||
| Apr 14, 2016 at 18:01 | answer | added | Fedor Petrov | timeline score: 4 | |
| Apr 14, 2016 at 17:57 | comment | added | David E Speyer | @MoritzFirsching The nonbases of nonrealizable matroids tend to have little overlap with each other, so there can't be that many of them. Matroids with few bases tend to involve lots of points living in much lower dimensional flats, which is pretty easy to realize. All of the examples mentioned in mathoverflow.net/questions/212411/… , for example, are easily realizable. (Of course, this is all intuition until we actually prove something.) | |
| Apr 14, 2016 at 17:53 | comment | added | Moritz Firsching | @DavidSpeyer: what makes you think that this would be realizable? | |
| Apr 14, 2016 at 17:34 | comment | added | David E Speyer | @FedorPetrov Right, but phrase it as "What is the minimal number of bases of a matroid of rank $d+1$ on $n$ points, where all circuits have size at least $4$?" (A circuit of size 1 would be the zero vector, so not a point of $\mathbb{P}^d$; a circuit of size 2 would be two parallel vectors, so the same point of $\mathbb{P}^d$ counted twice; a circuit of size 3 would give three colinear points in $\mathbb{P}^d$, so not the vertices of a polytope.) It is possible that the optimal answer to this question is not realizable as the vertices of a convex polytope, but I would bet otherwise. | |
| Apr 14, 2016 at 17:02 | comment | added | Fedor Petrov | Convexity is not matroidal property until you consider oriented matroid. And this still looks messy on first glance. | |
| Apr 14, 2016 at 16:25 | review | First posts | |||
| Apr 14, 2016 at 16:28 | |||||
| Apr 14, 2016 at 16:24 | history | asked | Mr Shorly | CC BY-SA 3.0 |