Rota's conjecture

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The objective of this Polymath project is to prove

Rota's Basis 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 [math]\displaystyle{ n }[/math] vectors in that column form a basis of [math]\displaystyle{ V }[/math].

Definitions

The statement of Rota's Basis Conjecture is elementary enough that definitions are not necessary, but we present here some definitions that are used below.

A matroid is a finite set [math]\displaystyle{ E }[/math] together with a family of subsets of [math]\displaystyle{ E }[/math] (called independent sets) such that

1. if [math]\displaystyle{ J }[/math] and [math]\displaystyle{ I }[/math][math]\displaystyle{ J }[/math] then [math]\displaystyle{ I }[/math], and
2. if [math]\displaystyle{ I, J }[/math] and [math]\displaystyle{ |I| < |J| }[/math] then there exists [math]\displaystyle{ x }[/math][math]\displaystyle{ J }[/math] such that [math]\displaystyle{ I ∪ {x} }[/math].

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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