# Rota, Gian-Carlo

## Rota's basis conjecture ★★★

Author(s): Rota

\begin{conjecture} Let $V$ be a vector space of dimension $n$ and let $B_1,\ldots,B_n \subseteq V$ be bases. Then there exist $n$ disjoint transversals of $B_1,\ldots,B_n$ each of which is a base. \end{conjecture}

Keywords: base; latin square; linear algebra; matroid; transversal

## Rota's unimodal conjecture ★★★

Author(s): Rota

Let $M$ be a matroid of rank $r$, and for $0 \le i \le r$ let $w_i$ be the number of closed sets of rank $i$.

\begin{conjecture} $w_0,w_1,\ldots,w_r$ is unimodal. \end{conjecture}

\begin{conjecture} $w_0,w_1,\ldots,w_r$ is log-concave. \end{conjecture}

Keywords: flat; log-concave; matroid