On the Lines–Planes Inequality for Matroids

✍ Scribed by Joseph P.S. Kung

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 101 KB
• Volume: 91
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2000.3101

✦ Synopsis

dedicated to the memory of gian-carlo rota

1. THE UNIMODALITY CONJECTURE

Although Gian-Carlo Rota did not publish much in matroid theory, his influence on the subject is pervasive (see ). Among the many conjectures bearing his name in matroid theory, the unimodality conjecture is perhaps the most intractable.

Let G be a combinatorial geometry (or simple matroid). The ith Whitney number W i (of the second kind ) is the number of rank-i flats in G. Thus, W 1 is the number of points, W 2 is the number of lines, and W 3 is the number of planes.

Rota's Unimodality Conjecture. Let G be a rank-n geometry. Then, the sequence W 0 , W 1 , W 2 , ..., W n is unimodal, that is, there is a rank s such that

One of Rota's motivation is that the Minkowski mixed volumes of a convex set form a unimodal sequence (see ). There might be a way to use methods or ideas from convexity theory to prove the unimodality conjecture.

In this paper, we present a partial result about the case n=5 of the unimodality conjecture.

1.1. Theorem. Let G be a geometry of rank at least 5 in which all the lines have the same number of points. Then W 2 W 3 .