Rota's conjecture: Difference between revisions
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* [AB2006] [http://www.ams.org/journals/tran/2006-358-11/S0002-9947-06-03833-5/S0002-9947-06-03833-5.pdf The intersection of a matroid and a simplicial complex], Ron Aharoni and Eli Burger, Trans. Amer. Math. Soc. 358 (2006), 4895-4917. | * [AB2006] [http://www.ams.org/journals/tran/2006-358-11/S0002-9947-06-03833-5/S0002-9947-06-03833-5.pdf The intersection of a matroid and a simplicial complex], Ron Aharoni and Eli Burger, Trans. Amer. Math. Soc. 358 (2006), 4895-4917. | ||
* [C1995] [http:// | * [C1995] [http://alum.mit.edu/www/tchow/dinitz.pdf On the Dinitz conjecture and related conjectures], Timothy Chow, Disc. Math. 145 (1995), 73-82. | ||
* [C2009] [http:// | * [C2009] [http://alum.mit.edu/www/tchow/rotathree.pdf Reduction of Rota's basis conjecture to a conjecture on three bases], Timothy Chow, Siam J. Disc. Math. 23 (2009), 369-371. | ||
* [EE2015] [https://arxiv.org/pdf/1502.03736v1.pdf Furstenberg sets and Furstenberg schemes over finite fields], Jordan Ellenberg, Daniel Erman, Feb 2015. | * [EE2015] [https://arxiv.org/pdf/1502.03736v1.pdf Furstenberg sets and Furstenberg schemes over finite fields], Jordan Ellenberg, Daniel Erman, Feb 2015. | ||
* [GH2006] [https://ir.canterbury.ac.nz/bitstream/handle/10092/11877/geelen_humphries_UCDMS2006-6_report.pdf?sequence=1&isAllowed=y Rota's basis conjecture for paving matroids], J. Geelen, P. Humphries, SIAM J. Discrete Math. 20 (2006), no. 4, 1042–1045. | * [GH2006] [https://ir.canterbury.ac.nz/bitstream/handle/10092/11877/geelen_humphries_UCDMS2006-6_report.pdf?sequence=1&isAllowed=y Rota's basis conjecture for paving matroids], J. Geelen, P. Humphries, SIAM J. Discrete Math. 20 (2006), no. 4, 1042–1045. | ||
Revision as of 12:17, 23 March 2017
The objective of this Polymath project is to prove
- Rota's conjecture: if [math]\displaystyle{ B_1,\dots,B_n }[/math] are [math]\displaystyle{ n }[/math] bases of an [math]\displaystyle{ n }[/math]-dimensional vector space [math]\displaystyle{ V }[/math] (not necessarily distinct or disjoint), then there exists an [math]\displaystyle{ n \times n }[/math] grid of vectors [math]\displaystyle{ (v_{ij}) }[/math] such that
- 1. the [math]\displaystyle{ n }[/math] vectors in row [math]\displaystyle{ i }[/math] are the members of the [math]\displaystyle{ i^{th} }[/math] basis [math]\displaystyle{ 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.
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.
