In 1989, 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 (for example, 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.

中文翻译：

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Rota 基本猜想的半途而废

1989年，罗塔做出了如下猜想。给定 $n$ 个基 $B_{1},\dots,B_{n}$ 在 $n$ 维向量空间 $V$ 中，我们总能找到 $V$ 的 $n$ 个不相交基，每个基包含一个来自每个 $B_{i}$ 的元素（我们称这样的基为横向基）。尽管 Rota 的基础猜想看起来很简单，并且经过了许多研究人员的努力（例如，该猜想最近成为合作“Polymath”项目的主题），但它仍然是开放的。在本文中，我们证明总是可以找到 $\left(1/2-o\left(1\right)\right)n$ 不相交的横向基，改进了之前的 $\Omega\left(n/ \log n\right)$。我们的结果也适用于拟阵的更一般设置。