If you think that there are no universals, then you're a nominalist aren't you?
No. Nominalism is an "explanation," of and one theory of universals. It doesn't deny them, it says they are only "names."
The whole questions of universals is very confusing because there are two distinctly different concepts, both called universals, and their difference is almost never noted. The Platonic version is ontological, and in some sense, never made explicit, they are thought of as actual ontological (metaphysical) real things. That is the basis of Platonic "realism." The other version of "universals" is epistemological. That is the version most people have in mind today, and the version Ayn Rand thought she was addressing. (Huemer, while explicitly denying Platonic realism, slips into it when saying things like qualities "actually being in more than one thing." Qualities aren't "in" things in that sense at all, they are "of" things.)
And that is the problem. The epistemological version is an attempt to make sense of the ontological mistake. It essentially says, "yes there are universals, but they are not actual metaphysical things, they are only concepts for [and this is the basis of all the different theories] something ...." But what? If you think they are only concepts which "name" the similarity in things you are a nominalist. (If you think they are only concepts for abstract ideas for similarities in things you are a conceptualist. If you went to a Catholic university, you probably believe universals do have ontological existence and are a realist. You might be surprised how prevalent this view still is.)
Both the nominalist and conceptualist, "solutions," however, are inadvertent denials of the reality of the very thing universals supposedly identify, the qualities (characteristics, attributes, and properties) of existents. But the concept is not needed at all. Things are whatever their qualities are. There is nothing else to know about anything except what their characteristics, attributes, and properties, that is, their qualities are.
There are qualities because there are existents. Qualities are "real" because existents are "real" and the identity of any existent is all it's qualities. Some existents have the same qualities as other existents. Period. No special concept is required by that fact. The concept of universals is just a mistake, and all attempts to explain them are exercises in futility. "Universals" is a pseudo-concept that needs to be purged, once-and-for-all, from philosophy.