Rota's unimodal conjecture

Importance: High ✭✭✭
Author(s): Rota, Gian-Carlo
Recomm. for undergrads: no
Posted by: mdevos
on: June 8th, 2007

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}

A sequence $a_0,a_1,\ldots a_n$ is \emph{log-concave} if $a_i^2 \ge a_{i-1} a_{i+1}$ for all $1 \le i \le n-1$.

The first of these conjectures is due to Rota [R], the second is folklore as far as I (M. DeVos) know. The special case of proving the second conjecture for $w_1,w_2,w_3$ amounts to showing that $(\#lines)^2 \ge (\#points)(\#planes)$ and has been called the points-lines-planes conjecture. Seymour [S] proved this conjecture in the special case where every line contains at most four points, but it is still open in general.


*[R] Rota, Gian-Carlo, Combinatorial theory, old and new. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, pp. 229--233. Gauthier-Villars, Paris, 1971. \MRhref{0505646}

[S] Seymour, P. D. On the points-lines-planes conjecture, J. Combin. Theory Ser. B 33 (1982), no. 1, 17--26. \MRhref{0678168}

* indicates original appearance(s) of problem.