Matroids with large branch-depth

Abstract

We prove a conjecture of DeVos, Kwon and Oum that matroids of high branch-depth have large uniform minors or large fan minors. The groundset of a large matroid of low branchdepth admits a partition into many sets such that the union of any subcollection has low connectivity. Most of the work in proving the conjecture goes into finding obstructions to finding such partitions. In particular, we prove that forbidding a given uniform matroid and a given fan as a minor guarantees the existence of such partitions in all sufficiently large matroids.