Every Graph is Essential to Large Treewidth

We show that for every graph $H$, there is a hereditary weakly sparse graph class ${\mathcal{C}}_H$ of unbounded treewidth such that the $H$-free (i.e., excluding $H$ as an induced subgraph) graphs of ${\mathcal{C}}_H$ have bounded treewidth. This refutes several conjectures and critically thwarts the quest for the unavoidable induced subgraphs in classes of unbounded treewidth, a wished-for counterpart of the Grid Minor theorem. We actually show a stronger result: For every positive integer $t$, there is a hereditary graph class ${\mathcal{C}}_t$ of unbounded treewidth such that for any graph $H$ of treewidth at most $t$, the $H$-free graphs of ${\mathcal{C}}_t$ have bounded treewidth. Our construction is a variant of so-called layered wheels. We also introduce a framework of abstract layered wheels, based on their most salient properties. In particular, we streamline and extend key lemmas previously shown on individual layered wheels. We believe that this should greatly help develop this topic, which appears to be a very strong yet underexploited source of counterexamples.