Curious Characterizations of Projective and Affine Geometries

✍ Scribed by Joseph P.S. Kung

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 136 KB
• Volume: 28
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.2001.0793

✦ Synopsis

We prove the following characterization theorem: If any three of the following four matroid invariants-the number of points, the number of lines, the coefficient of λ n-2 in the characteristic polynomial, and the number of three-element dependent sets-of a rank-n combinatorial geometry (or simple matroid) are the same as those of a rank-n projective geometry, then it is a projective geometry (of the same order).

To do this, we use a lemma which is of independent interest: If H is a geometry in which all the lines have exactly -1 or points and G is a geometry with at least three of the four matroid invariants the same as H, then all the lines in G also have exactly -1 or points. An analogue of the characterization theorem holds for affine geometries. Our methods also yield inequalities amongst the four matroid invariants.  2002 Elsevier Science (USA)