Curious Characterizations of Projective and Affine Geometries |
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Institution: | Department of Mathematics, University of North Texas, Denton, Texas, 76203 |
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Abstract: | 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 λn − 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 ℓ − 1 or ℓ 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 ℓ − 1 or ℓ points. An analogue of the characterization theorem holds for affine geometries. Our methods also yield inequalities amongst the four matroid invariants. |
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