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Halfway to Rota’s Basis Conjecture

Bucić, M; Kwan, M; Pokrovskiy, A; Sudakov, B; (2020) Halfway to Rota’s Basis Conjecture. International Mathematics Research Notices 10.1093/imrn/rnaa004. (In press). Green open access

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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/ log n). Our results also apply to the more general setting of matroids.

Type: Article
Title: Halfway to Rota’s Basis Conjecture
Open access status: An open access version is available from UCL Discovery
DOI: 10.1093/imrn/rnaa004
Publisher version: https://doi.org/10.1093/imrn/rnaa004
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10112648
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