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CPC

2000

2000

We examine the specialization to simple matroids of certain problems in extremal matroid theory that are concerned with bounded cocircuit size. Assume that each cocircuit of a simple matroid M has at most d elements. We show that if M has rank 3, then M has at most d + d + 1 points and we classify the rank-3 simple matroids M that have exactly d + d + 1 points. We show that if M is a connected matroid of rank 4 and d is q3 with q > 1, then M has at most q3 +q2 +q +1 points; this upper bound is strict unless q is a prime power, in which case the only such matroid with exactly q3 +q2 +q +1 points is the projective geometry PG(3, q). We also show that if d is q4 for a positive integer q and if M has rank 5 and is vertically 5-connected, then M has at most q4 + q3 + q2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case PG(4, q) is the only such matroid that attains this bound.

Added |
17 Dec 2010 |

Updated |
17 Dec 2010 |

Type |
Journal |

Year |
2000 |

Where |
CPC |

Authors |
Joseph E. Bonin, Talmage James Reid |

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