On a generalisation of spikes

TL;DR

This paper characterizes matroids with the $(t, ext{ell})$-property, showing finiteness for $ ext{ell}<2t$, and proves large matroids with the $(t,2t)$-property are $t$-spikes, revealing their structure and properties.

Contribution

It introduces the concept of $t$-spikes and proves that large matroids with the $(t,2t)$-property are necessarily $t$-spikes, extending understanding of matroid structure.

Findings

Finite number of matroids with $(t, ext{ell})$-property for $ ext{ell}<2t$.

Infinite family of matroids with $(t,2t)$-property.

Large $(t,2t)$-property matroids are $t$-spikes.

Abstract

We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for any positive integer $t$, there is a finite number of matroids with the $(t,\ell)$-property for $\ell<2t$; however, matroids with the $(t,2t)$-property form an infinite family. We say a matroid is a $t$-spike if there is a partition of the ground set into pairs such that the union of any $t$ pairs is a circuit and a cocircuit. Our main result is that if a sufficiently large matroid has the $(t,2t)$-property, then it is a $t$-spike. Finally, we present some properties of $t$-spikes.