A Note on the Sticky Matroid Conjecture

TL;DR

This paper investigates the properties of sticky matroids, showing that certain high-rank matroids with disjoint hyperplanes are not sticky, and connects the conjecture to a broader unresolved problem in matroid theory.

Contribution

It extends previous results by proving non-stickiness for rank-r matroids with disjoint hyperplanes for r ≥ 3, linking the sticky matroid conjecture to Kantor's conjecture.

Findings

No rank-3 matroid with two disjoint lines is sticky.

No rank-r matroid with two disjoint hyperplanes is sticky for r ≥ 3.

The sticky matroid conjecture depends on the resolution of Kantor's conjecture in rank 4.

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.