Odysseus uses flattery on Calypso's island by complimenting her over his wife Penelope. He makes her believe that the only reason he wants to be back home is because of Nostos, or homesickness. He bluntly states that Calypso is more beautiful and so on.
Odysseus flatters Calypso by expressing admiration for her beauty, intelligence, and powers as a goddess. He tells her that no mortal woman can compare to her and recognizes her as a superior being worthy of his attention and affection.
Calypso offers immortality to Odysseus
Odysseus met Calypso on her lush island of Ogygia.
Hermes tells Calypso to let Odysseus go.
Calypso
Calypso has the power of immorality over Odysseus.
Hermes is sent down to Calypso, telling her that the gods wish for her to release Odysseus. Therefore, it is Calypso who allows Odysseus to leave, not Odysseus who 'manages to say no.'
Calypso
Calypso and Odysseus become lovers. Calypso offers Odysseus immortality and eternal youth if he would stay with her. Calypso does not aid Odysseus' escape until told to do so, so he seemed trapped on her island without a ship.
Calypso wants Odysseus to stay with her on the island forever.
Calypso
No! Poseidon tries to drown Odysseus during his escape from Calypso. Athena and Hermes assist Odysseus.
Odysseus stays with Calypso [The Hider] on Ogygia for seven years.