@uofp2020
Princeton likes to think it is Yale and Harvard’s athletic rival but it is not since these schools only care about each other. There is a Yale saying that goes ”Harvard sucks, but Princeton doesn’t matter.”
Also Penn and Columbia like to think that Princeton is their rival but actually Princeton doesn’t care too much about either and thinks neither is a full-fledged rival. However, it does care about Penn a bit more (the Penn-Princeton basketball rivalry is actually pretty big) and if one had to choose from the two schools, Penn definitely is more of a rival of Princeton than Columbia. Still tho the word rival is a bit of a stretch since Penn people feel more strongly about the rivalry than Princeton people.
Just so that you can get an idea:
http://dailyprincetonian.com/opinion/2007/12/princeton-the-third-wheel/