Unavoidable induced subgraphs in graphs with complete bipartite induced minors

We prove that if a graph contains the complete bipartite graph $K_{134,12}$ as an induced minor, then it contains a cycle of length at most~12 or a theta as an induced subgraph. With a longer and more technical proof, we prove that if a graph contains $K_{3,4}$ as an induced minor, then it contains a triangle or a theta as an induced subgraph. Here, a \emph{theta} is a graph made of three internally vertex-disjoint chordless paths $P_1=a\dots b$, $P_2=a\dots b$, $P_3=a\dots b$, each of length at least two, such that no edges exist between the paths except the three edges incident to $a$ and the three edges incident to $b$. A consequence is that excluding a grid and a complete bipartite graph as induced minors is not enough to guarantee a bounded tree-independence number, or even that the treewidth is bounded by a function of the size of the maximum clique, because the existence of graphs with large treewidth that contain no triangles or thetas as induced subgraphs is already known (the so-called layered wheels).