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Halfway to Rota’s basis conjecture. (English) Zbl 1465.05026

Summary: In 1989, G.-C. Rota made the following conjecture. Given \(n\) bases \(B_1,\dots ,B_n\) in an \(n\)-dimensional vector space \(V\), one can always find \(n\) disjoint bases of \(V\), each containing exactly one element from each \(B_i\) (we call such bases transversal bases). Rota’s basis conjecture remains wide open despite its apparent simplicity and the efforts of many researchers (e.g., the conjecture was recently the subject of the collaborative “Polymath” project). In this paper we prove that one can always find \(\left (1/2-o\left (1\right )\right )n\) disjoint transversal bases, improving on the previous best bound of \(\Omega \left (n/\log n\right )\). Our results also apply to the more general setting of matroids.

MSC:

05B35 Combinatorial aspects of matroids and geometric lattices
15A03 Vector spaces, linear dependence, rank, lineability
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