In Queensland, pawpaws are supposed to ripen on the tree - they are nowadays packed in an expanded net of soft poly foam inside cardboard boxes to be taken to market.

Almost every family in the 1950-60's had a pawpaw tree in their backyard, or a relative's, so while they were in the shops, it was not till we moved to the city in the 70's and ours died that we started to buy them. We never ate green ones and never tried to cook them, as my grandma told me that you would get a tummy ache (c'mon, when I was 5, she knew everything about the world!). I have since discovered that green ones may be cooked, or even julienned and placed in salads. If one needed to be ripened a bit more, it would be placed on a windowsill or similar where it could sit in the sun for a few days, undisturbed.

The ripe (red - not granny-smith) apple trick we used for bananas - the apple gives off acetylene, confined in a brown paper bag, it used to bring them along nicely.

The seeds of the old non-hybridized ones are easily sprouted, then transplanted, often they would spring up in the (organic, but we didn't call it that then!) rubbush pile.