What customs does Odysseus make it clear that they are counting upon?
This is a great question. In book nine of the Odyssey, Odysseus and his men come upon the island of the Cyclopes. They are famished and in need of rest. As they explore the island a little, they come upon a cave where there are crates of milk and cheese and they are surprised. Odysseus' men bid him to leave, but Odysseus lingers for a while to his dismay, because he realizes that the owner of the cave is Polyphemus, a one-eyed monster.
It is very possible that Odysseus lingers, because he expects the hospitatlity of the inhabitants on the island. Within the Greek world one of the most important customs was hospitality (xenia - in Greek). Not to show hospitality was to be uncivilized and this is exactly how the inhabitants of the island are described. No wonder they know nothing of hospitality.
In light of this, Odysseus has to use his cunning to escape the island and he does.