On The Number Of Elements In Matroids With Small Circuits Or Cocircuits

It has been conjectured that a connected matroid with largest circuit size c ≥ 2 and largest cocircuit size c* ≥ 2 has at most ½cc* elements. Pou-Lin Wu has shown that this conjecture holds for graphic matroids. We prove two special cases of the conjecture, not restricted to graphic matroids, thereby providing the first nontrivial evidence that the conjecture is true for non-graphic matroids. Specifically, we prove the special case of the conjecture in which c = 4 or c* = 4. We also prove the special case for binary matroids with c = 5 or c* = 5.