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A Note on the Sticky Matroid Conjecture

Joseph E. Bonin

Abstract

A matroid is sticky if any two of its extensions by disjoint sets can be glued together along the common restriction (that is, they have an amalgam). The sticky matroid conjecture asserts that a matroid is sticky if and only if it is modular. Poljak and Turzik proved that no rank-3 matroid having two disjoint lines is sticky. We show that, for r at least 3, no rank-r matroid having two disjoint hyperplanes is sticky. These and earlier results show that the sticky matroid conjecture for finite matroids would follow from a positive resolution of the rank-4 case of a conjecture of Kantor.

A Note on the Sticky Matroid Conjecture

Abstract

A matroid is sticky if any two of its extensions by disjoint sets can be glued together along the common restriction (that is, they have an amalgam). The sticky matroid conjecture asserts that a matroid is sticky if and only if it is modular. Poljak and Turzik proved that no rank-3 matroid having two disjoint lines is sticky. We show that, for r at least 3, no rank-r matroid having two disjoint hyperplanes is sticky. These and earlier results show that the sticky matroid conjecture for finite matroids would follow from a positive resolution of the rank-4 case of a conjecture of Kantor.

Paper Structure

This paper contains 3 sections, 4 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Results

Key Result

Theorem 2.1

Let $\mathcal{Z}$ be a collection of subsets of a set $S$ and let $r$ be an integer-valued function on $\mathcal{Z}$. There is a matroid for which $\mathcal{Z}$ is the collection of cyclic flats and $r$ is the rank function restricted to the sets in $\mathcal{Z}$ if and only if

Figures (2)

  • Figure 1: The Vámos matroid.
  • Figure 2: The lattice $\mathcal{Z}(N)$ in the proof of Theorem \ref{['thm:planes']}.

Theorems & Definitions (7)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof