Rand claimed to have solved the "problem of universals." But it is not clear that she even understood this problem. Rand frames the problem as follows:
The issue of concepts (known as "the problem of universals") is philosophy's central issue. Since man's knowledge is gained and held in conceptual form, the validity of man's knowledge depends on the validity of concepts. But concepts are abstractions or universals, and everything that man perceives is particular, concrete. What is the relationship between abstractions and concretes? To what precisely do concepts refer in reality? Do they refer to something real, something that exist--or are they merely inventions of man's mind, arbitrary constructs or loose approximations that cannot claim to represent knowledge?
Now contrast that with how the issue is framed on Wikipedia:
The problem of universals is an ancient problem in metaphysics about whether universals exist. Universals are general or abstract qualities, characteristics, properties, kinds or relations, such as being male/female, solid/liquid/gas, or a certain colour, that can be predicated of individuals or particulars, or that individuals or particulars can be regarded as sharing or participating in. For example, Scott, Pat, and Chris have in common the universal quality of being human or humanity. While many standard cases of universals are also typically regarded as abstract objects (such as humanity), abstract objects are not necessarily universals. For example, numbers can be held to be particular yet abstract objects.
Rand's version of the problem of universals is different from Wikipedia's. Now I have no interest in getting into an argument over which version is "correct." Wikipedia is merely relating how the mediavals originally framed the problem. Rand was far more interested in the epistemological side of the issue. In fact, she pretty much ignores the metaphysical issue — so much so that it is not always clear how to characterize Rand's position on universals. Without explicitly saying so, Rand seems to reject the view that universals exist in
reality. Instead, she seems to adopt a version of conceptualism. Only particulars exist in reality; but concepts (which for Rand are identical with universals) correspond "objectively" to multiple instances of particulars in reality. But how do they do so? Rand replies: A concept refers to a group of particulars in reality possessing characteristics in common which differ only in their measurements.
This, however, is rather vague. Rand attacked nominalism for regarding "abstractions" as mere "'names' which we give to arbitrary groupings of concretes on the basis of vague resemblances." [IOTE, 2] But why isn't Rand's own theory guilty of the same charge? What makes her theory "objective" and nominalism "arbitrary"? Rand never explains, she merely asserts.