Unavoidable pivot-minors in graphs of large rank-depth

Shrub-depth and rank-depth are related graph parameters that are dense analogs of tree-depth. We prove that for every positive integer $t$, every graph of sufficiently large rank-depth contains a pivot-minor isomorphic to a path on $t$ vertices or a graph consisting of two disjoint cliques of size $t$ joined by a half graph. This answers an open problem raised by Kwon, McCarty, Oum, and Wollan in 2021.