On finite matroids with two more hyperplanes than points. (English) Zbl 1083.05011
Summary: One of the most interesting results about finite matroids of finite rank and generalized projective spaces is the result of J. G. Basterfield and L. M. Kelly [Proc. Camb. Philos. Soc. 64, 585–588 (1968; Zbl 0183.49602)] and of C. Greene [J. Comb. Theory 9, 357–364 (1970; Zbl 0254.05016)] affirming that any matroid contains at least as many hyperplanes as points, with equality in the case of generalized projective spaces. Consequently, the goal is to characterize and classify all matroids containing more hyperplanes than points. In 1996, I obtained the classification of all finite matroids containing one more hyperplane than points. In this paper a complete classification of finite matroids with two more hyperplanes than points is obtained. Moreover, a partial contribution to the classification of those matroids containing a certain number of hyperplanes more than points is presented.
MSC:
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 51E15 | Finite affine and projective planes (geometric aspects) |
References:
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