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Halin proved that every minimally $k$-connected graph has a vertex of degree$k$. More generally, does every minimally vertically $k$-connected matroid havea $k$-element cocircuit? Results of Murty and Wong give an affirmative answerwhen $k \le 3$. We show that every minimally vertically $4$-connected matroidwith at least six elements has a $4$-element cocircuit, or a $5$-elementcocircuit that contains a triangle, with the exception of a specific non-binary$9$-element matroid. Consequently, every minimally vertically $4$-connectedbinary matroid with at least six elements has a $4$-element cocircuit.

Small cocircuits in minimally vertically $4$-connected matroids