Lemma 1.21. Assuming that:

$A \subseteq G$ is finite
$E (A) \geq η | A |^{3}$
$c > 0$

Then there is a subset

X \subseteq A

of size

| X | \geq η | A | ∕ 3

such that for all but a

(16 c)

-proportion of pairs

(a, b) \in X^{2}

,

a - b \in P_{c η}

.