Let be a matroid on an -element set that is a disjoint union of independent sets of size . Assume that there exists another matroid on the same ground set with the following properties:

(1) is strongly base orderable.

(2) for all , where is the rank function of .

(3) All circuits of satisfying remain dependent in .

Then there is an grid whose th row comprises and whose columns are independent in .

That all strongly base-orderable matroids satisfy Rota’s Basis Conjecture follows at once if we take since condition (2) is automatic by pigeonhole. However, there is a lot more that we might extract out of Lemma 6. Wild asks whether a suitable *always* exists. As Wild recognizes, this is probably too optimistic, but he doesn’t have a counterexample. Maybe a suitable always exists for graphic matroids?

I expect that the answer is probably no, but the counterexamples should be more interesting, because I don’t think you can get them just by stringing together disjoint copies of and “uncontracting” edges as in Luke Pebody’s example.

Should the answer unexpectedly be yes, then that gives hope that we can control the copies of . Pulling out edges can be thought of as a proxy for making sure that those edges get used in other columns, in some kind of induction argument.

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