The five-vertex path $P_5$ has the Erdős–Hajnal property: there is a positive integer $q$ such that every finite graph $G$ with no induced copy of $P_5$ satisfies

For every finite graph $H$ and every positive integer $E$, there is a positive integer $D$ such that every finite induced-$H$-free graph $G$ contains a set $X$ satisfying $|V(G)|\leq D|X|$ and such that either $G[X]$ or its complement has edge density at most $1/E$. This is the cleared-denominator finite form of Rödl's theorem cited as Theorem 4.1 in the paper.

This is the denominator-cleared form of Lemma 6.2 of Nguyen, Scott, and Seymour. A sufficiently large house-free graph contains $E$ disjoint polynomially large blocks, and every pair of blocks is either complete or has edge density at most $E^{-d}$.

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.

This denominator-cleared form of Lemma 4.1 groups the connected components of the complement. Either one anticonnected component has size at least $|T|/Q^2$, or there are $Q$ pairwise complete groups, each of size at least $|T|/(4Q^3)$.

The $d=1/2$ case of Theorem 2.1, with denominators cleared. Let $A$ and $B$ be disjoint vertex sets. Suppose that every vertex of $B$ has a neighbor in $A$ and that every vertex of $A$ has at most $\Delta$ neighbors in $B$. For every $\Gamma>0$, either there is a comb with $t\geq 1$ teeth whose blocks satisfy $\Gamma\leq t^2|B_i|$, or $|B|^2\leq 128^2\Gamma\Delta$.

The constant $128$ is an absolute constant obtained from an integral four-adic form of the peeling argument. Its precise value is not used later.

A $1/32$-sparse $P_5$-free graph on a set $S$ of at least two vertices has two anticomplete sets, each of size at least $|S|/32$. This is Lemma 4.4 of Nguyen, Scott, and Seymour.

This is the denominator-cleared form of Lemma 6.2 of Nguyen, Scott, and Seymour. A sufficiently large house-free graph contains $E$ disjoint polynomially large blocks, and every pair of blocks is either complete or has edge density at most $E^{-d}$.

This is a denominator-cleared form of Lemma 7.1 of Nguyen, Scott, and Seymour. At scale $Q$, a sparse house-free graph either becomes polynomially sparser on a polynomial fraction of its vertices, contains a long complete blockade, or admits an anticomplete peel. In the last alternative, the set $Y$ omits at most a $3/Q$ fraction of the current vertex set.

This is the denominator-cleared preparation carried out in Claim 7.1.1 of Nguyen, Scott, and Seymour. Starting from the semisparse blockade of Lemma 6.2, one samples equal-sized subblocks, removes high cross-degree vertices, and takes large anticonnected components. Failure to find such a component already gives the complete-blockade alternative.

This finite greedy lemma is the sampling-and-cleaning step in Claim 7.1.1. Subblocks are selected in order so that every weakly sparse pair has a controlled edge count. Removing vertices of excessive cross-degree then gives vertexwise sparsity in both directions while retaining at least half of each sample.

For some integer $d\geq 2$, a house-free graph that is $1/R^2$-sparse on a set $S$, where $R\geq 64$, either becomes $1/R^{2d}$-sparse on a subset of size at least $|S|/R^{32d^3}$, or has a complete or anticomplete $R$-blockade whose blocks have size at least $|S|/R^{36d^3}$. This is a reciprocal-square form of Lemma 7.2 of Nguyen, Scott, and Seymour.

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.