There is an integer $a\geq 1$ such that, for every $E\geq 3$, every finite house-free graph $G$ has one of two outcomes. Either an induced subgraph on $X$ is $1/E$-restricted and $|G|\leq E^a|X|$, or $G$ has a complete or anticomplete blockade of length $k$, where $2\leq k\leq E$, whose blocks all have size at least $|G|/k^a$.

This is the denominator-cleared form of Lemma 7.3 of Nguyen, Scott, and Seymour. The structural argument is stated for the house, the complement of the five-vertex path.