Open Collections

Description

In 1989, Rota made the following conjecture. Given $n$ bases $B_1,\ldots, B_n$ 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). Rotaâ s basis conjecture remains open despite its apparent simplicity and the efforts of many researchers (for example, the conjecture was recently the subject of the collaborative â Polymathâ project). In this talk, I will discuss how to find $(0.5 - o(1))n$ disjoint transversal bases, improving the previously best known bound of $n/\log n$. This is joint work with Bucic, Kwan, and Sudakov.