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.

翻译：我们证明，对于任意图$H$，都存在一个无界树宽的遗传弱稀疏图类$\mathcal C_H$，使得$\mathcal C_H$中不含$H$（即排除$H$作为诱导子图）的图具有有界树宽。这一结果否定了若干猜想，并严重阻碍了在无界树宽图类中寻找不可避免的诱导子图的研究，而后者是网格子图定理的理想对应物。我们实际上证明了一个更强的结果：对于任意正整数$t$，都存在一个无界树宽的遗传图类$\mathcal C_t$，使得对于任意树宽不超过$t$的图$H$，$\mathcal C_t$中不含$H$的图都具有有界树宽。我们的构造是所谓分层轮图的一种变体。我们还引入了一个基于分层轮图最显著性质的抽象分层轮图框架。特别地，我们简化并扩展了先前针对单个分层轮图证明的关键引理。我们相信这将极大地推动该主题的发展，该主题似乎是一个极其强大但尚未被充分挖掘的反例来源。