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 A1,…,Am of the ground set there exists a set S ε C meeting each Ai 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 A1,…,Am of the ground set there exists a set S ε D meeting each Ai 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 Ai in at least (formula presented) elements.
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ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics