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Hindrance from a wasteful common independent set

Attila Joó

Abstract

For (potentially infinite) matroids $ M $ and $ N $, an $ (M,N) $-hindrance is a set $ H$ that is independent but not spanning in $ N.\mathsf{span}_M(H) $. This concept was introduced by Aharoni and Ziv in the very first paper investigating Nash-Williams' Matroid Intersection Conjecture. They proved that the conjecture is equivalent to the statement that the non-existence of hindrances implies the existence of an $ M $-independent spanning set of $ N $. In this paper we present a breakthrough towards the Matroid Intersection Conjecture. Namely, we found a matroidal generalization of the `popular vertex' approach applied in the proof of the infinite version of König's theorem. The main result of this paper is an application of this new approach to show that if $ M $ and $ N $ admit a common independent set $ I $ that is ``wasteful'' in the sense that $ r(M/I)<r(N/I) $, then there exists an $ (M,N) $-hindrance.

Hindrance from a wasteful common independent set

Abstract

For (potentially infinite) matroids and , an -hindrance is a set that is independent but not spanning in . This concept was introduced by Aharoni and Ziv in the very first paper investigating Nash-Williams' Matroid Intersection Conjecture. They proved that the conjecture is equivalent to the statement that the non-existence of hindrances implies the existence of an -independent spanning set of . In this paper we present a breakthrough towards the Matroid Intersection Conjecture. Namely, we found a matroidal generalization of the `popular vertex' approach applied in the proof of the infinite version of König's theorem. The main result of this paper is an application of this new approach to show that if and admit a common independent set that is ``wasteful'' in the sense that , then there exists an -hindrance.
Paper Structure (14 sections, 15 theorems, 1 equation)

This paper contains 14 sections, 15 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Set theory
  4. Digraphs
  5. Matroids
  6. Preparations
  7. The changes of a fundamental circuit
  8. A key lemma
  9. A new auxiliary digraph
  10. Proof of Theorem \ref{['thm: main']}
  11. Preprocessing
  12. The popularity matroid
  13. Bad edges
  14. Construction of a hindrance

Key Result

Theorem 1.2

If $M$ and $N$ are finitary matroids on a common ground set and there exists an $I \in \mathcal{I}(M)\cap \mathcal{I}(N)$ with $r(M/I)<r(N/I)$, then $(M,N)$ is hindered.

Theorems & Definitions (55)

  • Conjecture 1.1: aharoni1998intersection
  • Theorem 1.2
  • Lemma 2.2: Fodor's lemma, jech2002set
  • Corollary 2.3
  • proof
  • Claim 2.4
  • proof
  • Lemma 2.5
  • proof
  • proof
  • ...and 45 more