## Abstract

We consider matroids with the property that every subset of the ground set of size t is contained in both an ℓ-element circuit and an ℓ-element cocircuit; we say that such a matroid has the (t, ℓ)-property. We show that for any positive integer t, there is a finite number of matroids with the (t, ℓ)-property for ℓ < 2t; however, matroids with the (t, 2t)-property form an infinite family. We say a matroid is a t-spike if there is a partition of the ground set into pairs such that the union of any t pairs is a circuit and a cocircuit. Our main result is that if a sufficiently large matroid has the (t, 2t)-property, then it is a t-spike. Finally, we present some properties of t-spikes.

Original language | English |
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Pages (from-to) | 358-372 |

Number of pages | 15 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 33 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

## Keywords

- Circuit
- Cocircuit
- Matroid
- Spike