Generalized spikes with circuits and cocircuits of different cardinalities

TL;DR

This paper characterizes large matroids with specific circuit and cocircuit containment properties, showing they are essentially generalized spikes, extending previous results where the parameters were equal.

Contribution

It generalizes the 2019 result by proving that sufficiently large matroids with the (s,2s,t,2t)-property are (s,t)-spikes, broadening understanding of matroid structure.

Findings

Large matroids with the (s,2s,t,2t)-property are (s,t)-spikes.

The paper extends previous results from the case s=t to more general parameters.

Properties of (s,t)-spikes are discussed.

Abstract

We consider matroids with the property that every subset of the ground set of size $s$ is contained in a $2s$-element circuit and every subset of size $t$ is contained in a $2t$-element cocircuit. We say that such a matroid has the \emph{$(s,2s,t,2t)$-property}. A matroid is an \emph{$(s,t)$-spike} if there is a partition of the ground set into pairs such that the union of any $s$ pairs is a circuit and the union of any $t$ pairs is a cocircuit. Our main result is that all sufficiently large matroids with the $(s,2s,t,2t)$-property are $(s,t)$-spikes, generalizing a 2019 result that proved the case where $s=t$. We also present some properties of $(s,t)$-spikes.