We show that every graph with pathwidth strictly less than $a$ that contains no path on $2^b$ vertices as a subgraph has treedepth at most $10ab$. The bound is best possible up to a constant factor.

翻译：我们证明，每个路径宽度严格小于 $a$ 且不包含 $2^b$ 个顶点的路径作为子图的图，其树深至多为 $10ab$。该上界在常数因子意义下是最优的。