ed by $B$.

In Section \[selfdualsection\], we prove a result of independent interest, Theorem \[selfdual\], that finds the unavoidable minors for arbitrary large matroids that have two disjoint bases. A corollary is the following, which finds one of two specific minors in any matroid that is not close to being ‘trivial’.

\[unavoidable\] Let $s \ge 0$ be an integer and $k = 4^{4^{2s^2}}$. Then, for each matroid $M$, either

-   $M$ has a $U_{s,2s}$-minor,

-   $M$ has a minor isomorphic to the direct sum of $s$ copies of $U_{1,2}$, or

-   there is a distance-$k$ perturbation of $M$ whose elements are all loops or coloops.

Structure Theory {#structure-theory .unnumbered}
----------------

Theorems \[main1\] and \[main2\] fit into a larger, mostly conjectural, regime of structure theory in minor-closed classes omitting a uniform matroid. The first of these conjectures predicts the unavoidable minors for very highly connected matroids. A matroid is *vertically $k$-connected* if, for every $A \subseteq E(M)$ with $\lambda_M(A) < k-1$, either $A$ or $E(M)-A$ is spanning in $M$. The following conjecture was posed in \[\[highlyconnected\]\].

\[highconn\] For all $n \ge 2$ there is an integer $k$ such that, if $M$ is a vertically $k$-connected matroid with $|M|\ge 2k$, then $M$ or $M^*$ has a minor isomorphic to one of $M(K_n),B(K_n),$ or $U_{n,2n}$.

While $M(K_n)^*$ and $B(K_n)^*$ are not even vertically $4$-connected themselves, they do contain minors with high vertical connectivity; indeed, for each $k$ there is a graph $G$ so that $M(G)^*$ and $B(G)^*$ are both vertically $k$-connected. To obtain such a graph one can take a $k$-regular Cayley graph with girth at least $k$ (see Margulis \[\[margulis\]\] for the construction); by \[\[gr\], Theorem 3.4.2\], these graphs are $k$-connected.

In any case, the dual outcomes in Conjecture \[highconn\] are perhaps not needed if $M$ has large co-rank.

For all $n \ge 2$ there is an integer $k$ so that, if $M$ is a vertically $k$-connected matroid with $|M|\ge 