Papaya is one of my favorite fruits.
when I was working in the Philippines we had four big trees that my housegirl took care of for me to get the most and biggest fruit.

Leave it on the tree as long as possible, then pick it and let it sit another day till it's ultra ripe. Then in the ref to chill it. Mmmmm... almost as good as ice cream to me.

As another poster said, get a recipe for som tam.....its delicious.