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Base partition for mixed families of finitary and cofinitary matroids

Joshua Erde, Pascal Gollin, Attila Joó, Paul Knappe, Max Pitz

Abstract

Let ${\mathcal{M} = (M_i \colon i\in K)}$ be a finite or infinite family consisting of matroids on a common ground set $E$ each of which may be finitary or cofinitary. We prove the following Cantor-Bernstein-type result: If there is a collection of bases, one for each $M_i$, which covers the set $E$, and also a collection of bases which is pairwise disjoint, then there is a collection of bases which partitions $E$. We also show that the failure of this Cantor-Bernstein-type statement for arbitrary matroid families is consistent relative to the axioms of set theory ZFC.

Base partition for mixed families of finitary and cofinitary matroids

Abstract

Let be a finite or infinite family consisting of matroids on a common ground set each of which may be finitary or cofinitary. We prove the following Cantor-Bernstein-type result: If there is a collection of bases, one for each , which covers the set , and also a collection of bases which is pairwise disjoint, then there is a collection of bases which partitions . We also show that the failure of this Cantor-Bernstein-type statement for arbitrary matroid families is consistent relative to the axioms of set theory ZFC.

Paper Structure

This paper contains 13 sections, 19 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Open questions
  4. Structure of this paper
  5. Sketch of the proof
  6. Infinite Matroids
  7. Preparatory Lemmas
  8. Augmenting paths
  9. Tight sets
  10. Feasible extensions
  11. The main lemmas
  12. Proof of the main result
  13. A consistency result

Key Result

Theorem 1.1

Let $\lambda$ be an infinite cardinal. Then a graph admits a $\lambda$-partitioning if and only if it admits both a $\lambda$-packing and a $\lambda$-covering.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 28 more