How we cite our quotes:
(Menelaos:) ‘[…] and Aias would have escaped his doom, though Athene hated him, had he not gone wildly mad and tossed out a word of defiance; for he said that in despite of the gods he escaped the great gulf of the sea, and Poseidon heard him, loudly vaunting, and at once with his ponderous hands catching up the trident he drove it against the Gyrean rock, and split a piece off it, and part of it stayed where it was, but a splinter crashed in the water, and this was where Aias had been perched when he raved so madly. It carried him down to the depths of the endless and tossing main sea. So Aias died, when he had swallowed down the salt water.’ (4.502-511)
Aias’s story can be seen as a warning to Odysseus not to let his own pride get out of hand, lest he anger the gods with his hubris.
(Alkinoös:) 'Now let us go outside and make our endeavor in all contests, so that our stranger can tell his friends, after he reaches his home, by how much we surpass all others in boxing, wrestling, leaping and speed of our feet for running.' (8.100-103)
Alkinoös is pretty proud of his kingdom, but notice that he's not comparing them to gods, or anything—just other men. Still, this is one of those tricky moments where we're not sure if this is justifiable pride or simply boasting. Given that Homer seems to like the Phaiakians, we're pretty sure that it's justifiable.
(Odysseus:) 'I know well how to handle the polished bow, and would be first to strike any man with an arrow aimed at a company of hostile men … But I will say that I stand far out ahead of all others such as are living mortals now and feed on the earth. Only I will not set myself against men of the generations before, not with Herakles nor Eurytos of Oichalia, who set themselves against the immortals with the bow, and therefore great Eurytos died suddenly nor came to an old age in his own mansions, since Apollo in anger against him killed him, because he had challenged Apollo in archery.' (8.215-228)
It looks like Odysseus has learned his lesson since boasting to the Cyclops—at least, part of his lesson. He's still claiming to be the best archer living, but key word living. He's going to claim that he's the best archer ever or anything. Because that would just be bragging.