## Abstract

Mysteriously, hypergraphs that are the intersection of two matroids behave in some respects almost as well as one matroid. In the present paper we study one such phenomenon - the surprising ability of the intersection of two matroids to fairly represent the parts of a given partition of the ground set. For a simplicial complex C denote by β(C) the minimal number of edges from C needed to cover the ground set. If C is a matroid then for every partition A_{1},…,A_{m} of the ground set there exists a set S ε C meeting each A_{i} in at least (formula presented) elements. We conjecture that a slightly weaker result is true for the intersection of two matroids: if D = P ∩ Q, where P,Q are matroids on the same ground set V and (formula presented), then for every partition A_{1},…,Am of the ground set there exists a set S ε D meeting each A_{i} in at least (formula presented) elements. We prove that if m = 2 (meaning that the partition is into two sets) there is a set belonging to D meeting each A_{i} in at least (formula presented) elements.

Original language | English |
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Article number | #P4.10 |

Journal | Electronic Journal of Combinatorics |

Volume | 24 |

Issue number | 4 |

DOIs | |

State | Published - 6 Oct 2017 |

### Bibliographical note

Publisher Copyright:© 2017, Australian National University. All rights reserved.

## Keywords

- Dimatroid
- Fair representation

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics