The five-cycle has the Erdős–Hajnal property. Equivalently, there is a positive integer $q$ such that every finite graph $G$ with no induced five-cycle satisfies $|V(G)|\leq \max\{\alpha(G),\omega(G)\}^q$.

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.

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.

Lemma 3.1 of the paper, with reciprocal integer parameters. Let $K=2^{20}$. For every sufficiently large $E$, every $D>0$, and every lower bound $Q$, there is an exponent $q\geq\max\{3,Q\}$ with the following property. If $G$ is $q$-critical, $|V(G)|\leq D|X|$, and $E\deg_{G[X]}(x)<|X|$ for every $x\in X$, then $G$ contains a stable hubbed comb such that $E\leq Kt$ and $E|X|\leq Kt^2|B_i|$ for every block $B_i$.

The absolute constant $K$ replaces the constant $400$ in the paper to accommodate the integral form of Theorem 2.1. Its value does not affect the Erdős–Hajnal conclusion.

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.

Lemma 4.2 of the paper, with denominators cleared. Let $Z$ induce a graph of edge density at most $1/E$, and suppose $2m \leq |Z|+1$. Then $Z$ contains an $m$-element set $X$ such that $E\deg_{G[X]}(x) \leq 4(m-1)$ for every $x\in X$. The factor $4$ results from first selecting $2m-1$ vertices by averaging and then retaining $m$ vertices of low degree.

Lemma 4.3 of the paper, with denominators cleared. For every finite graph $H$ and every $E>0$, 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, either in $G[X]$ or in its complement, $E\deg(x)<|X|$ for every $x\in X$.

The five-cycle has the Erdős–Hajnal property. Equivalently, there is a positive integer $q$ such that every finite graph $G$ with no induced five-cycle satisfies $|V(G)|\leq \max\{\alpha(G),\omega(G)\}^q$.

This statement combines Lemmas 3.1 and 4.3 with the choice of parameters in the proof of Theorem 4.4. There are integers $q\geq 3$ and $A\leq 2^{q-2}$ such that every $q$-critical induced-$C_5$-free graph $G$ contains, either in $G$ or in its complement, a stable hubbed comb with $t\geq 2$ teeth and $|V(G)|\leq At^2|B_i|$ for every block $B_i$. The complementary alternative is required because Lemma 4.3 gives low maximum degree in either $G$ or its complement; both induced-$C_5$-freeness and $q$-criticality are invariant under complementation.