Halfway to Rota's basis conjecture

Bucić, M. and Kwan, M. and Pokrovskiy, Alexey and Sudakov, B. (2020) Halfway to Rota's basis conjecture. International Mathematics Research Notices 2020 (21), pp. 8007-8026. ISSN 1073-7928.

Abstract

In 1989, Rota made the following conjecture. Given n bases B1,…,Bn in an n-dimensional vector space V, one can always find n disjoint bases of V, each containing exactly one element from each Bi (we call such bases transversal bases). Rota's basis conjecture remains wide open despite its apparent simplicity and the efforts of many researchers (for example, the conjecture was recently the subject of the collaborative "Polymath" project). In this paper we prove that one can always find (1/2−o(1))n disjoint transversal bases, improving on the previous best bound of Ω(n/logn). Our results also apply to the more general setting of matroids

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