Rota's conjecture
From Polymath1Wiki
The objective of this Polymath project is to prove
- Rota's conjecture: if [math]B_1,\dots,B_n[/math] are [math]n[/math] bases of an [math]n[/math]-dimensional vector space [math]V[/math] (not necessarily distinct or disjoint), then there exists an [math]n \times n[/math] grid of vectors [math](v_{ij})[/math] such that
- 1. the [math]n[/math] vectors in row [math]i[/math] are the members of the [math]i^{th}[/math] basis [math]B_i[/math] (in some order), and
- 2. in each column of the matrix, the n vectors in that column form a basis of V.
Contents
Definitions
Partial results
Variants of the problem
Discussion
- Proposal on MathOverflow (Feb 15, 2016)
- Rota’s Basis Conjecture: Polymath 12? (Feb 23, 2017)
References
- [AB2006] The intersection of a matroid and a simplicial complex, Ron Aharoni and Eli Burger, Trans. Amer. Math. Soc. 358 (2006), 4895-4917.
- [C1995] On the Dinitz conjecture and related conjectures, Timothy Chow, Disc. Math. 145 (1995), 73-82.
- [C2009] Reduction of Rota's basis conjecture to a conjecture on three bases, Timothy Chow, Siam J. Disc. Math. 23 (2009), 369-371.
- [EE2015] Furstenberg sets and Furstenberg schemes over finite fields, Jordan Ellenberg, Daniel Erman, Feb 2015.
- [GH2006] Rota's basis conjecture for paving matroids, J. Geelen, P. Humphries, SIAM J. Discrete Math. 20 (2006), no. 4, 1042–1045.
- [HKL2010] On disjoint common bases in two matroids, Nicholas J. A. Harvey, Tam´as Kir´aly, and Lap Chi Lau, TR-2010-10. Published by the Egerv´ary Research Group, P´azm´any P. s´et´any 1/C, H–1117, Budapest, Hungary.