Generalized spikes with circuits and cocircuits of different cardinalities

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 (s,2s,t,2t)-property. A matroid is an (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.