Aha, that thread was mine!
https://www.alternatehistory.com/discussion/showthread.php?t=356796
Just discussing copying the Library of Alexandria.
It will cover that there were a number of libraries that were personal prestige projects.
Another interesting point is what is a university? There is something (I forget what exactly) that suggests that the Library was simply part of a 'university' complex, but whether we would call it a true university is up for debate.
The legions were seen as the best way to propagate schooling, but that they'd potentially develop other schools besides military tactics and military engineering. The obvious one being field medicine, and medicine in general.
A university, those would be very sparse, I thought 6 was being conservative and got quite summarily corrected. You may be able to have 3, perhaps even under Augustus, but they'd have to be wealthy cities, and stable, as the university would be one of the first things to be burnt in a tax revolt.