You could say there’s an implied verb and possibly an implied subject: “[These are] cozy styles to give or keep.” If you want “styles” to be the subject, though, there’s no particular verb implied for it. The infinitive phrase “to give or keep” serves as another adjective to “styles”.
Example:
“I have a hungry dog to feed.” (That’s a sentence, with the noun “dog” modified by “hungry” and “to feed”.)
Yeah, to make a complete sentence, it needs to be something like “Cozy styles to give or keep are all the rage this holiday season.” Or “Here are some free knitting patterns of cozy styles to give or keep.” “Give or keep” is not a predicate in your case. It’s more like an infinitive adjective phrase, further modifying your subject “styles.”
“styles” is not doing the giving or keeping. Therefore it is not a subject. It is an object, the thing being given or kept. (I think. It’s not entirely clear from the sentence fragment.)
Try it this way:
You can give or keep these cozy styles.
You - the subject - the person doing the giving or keeping
can give/can keep - the verb, the actions that are possible for the subject to take
these cozy styles - the objects, the things that you can give or keep
I can’t quote the rule, but I think the infinitive verb cannot stand alone as a predicate. If the infinitive is used, it needs to either have an auxiliary verb to support it, or act as a gerundive noun. The infinitive of a verb is the simple form preceded by “to”.
An infinitive is a form of a verb, but it can’t function as the main verb of a sentence. It doesn’t have a subject; it hasn’t been conjugated-- as its name implies, it isn’t in any tense.
You could even say “Cozy styles give or keep,” and have a grammatically complete sentence, although it doesn’t make much sense. The verbs “give” and “keep” both have meanings for which they are intransitive, but probably not the ones you intend.
Now, you don’t say what this is for. If it’s some kind of slogan, it doesn’t necessarily need to be a complete sentence. “Always loyal” isn’t a complete sentence. “Must see TV” isn’t a complete sentence. If you are looking for a slogan for something, you can use a fragment. Some people might decide to point it out to you, but I doubt anyone is going to say “I refuse to shop there-- they use sentence fragments as slogans.”