Lots of college students don’t use bed frames, but if the mattress is directly on the floor it’s going to get moldy.
It still will with a box spring, but college students tend to move every year which helps and you’d definitely notice if it is when moving.
What would be even better is a “slat frame” that doesn’t need a box spring and lifts it off the ground. You can get good metal ones on Amazon for cheap. Probably pretty close to local price for a box spring.