Rota's conjecture

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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.

Definitions

Partial results

Variants of the problem

Discussion

References

  • [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.

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