Define a Russell socks set as a countable set of (pairwise disjoint) pairs such that no infinite subset has a choice function. Of course, if ZFC is consistent then it proves that no such set exists (as the axiom of choice is precisely the statement that every set has a choice function). On the other hand, it is known to be consistent with ZF that such a set exists.
Many wonderful and entertaining consequences of such a set existing in a model of ZF can be found in papers such as 'On the number of Russell's socks [...]' by Herrlich and Tachtsis or in Ethan Thomas' undergraduate thesis on the subject.
Neither of these papers explicitly construct a model of ZF containing a Russell socks set. Are any of the more common models such as Cohen's known to contain such a set? Is it easy to construct a model containing one?
Any reference would be much appreciated!
Yes. Cohen's second model of $\lnot\sf AC$ is a model in which there is a Russell set.
The proof can be found in Jech, "The Axiom of Choice" in Chapter 5, section 4. While Jech does not include the statement that the resulting set is a Russell set, it is implicit in the proof of Lemma 5.19.
Additionally, Fraenkel's second model of $\sf ZFA$ has a Russell set, and in the same book by Jech, he provides "transfer theorems" for transferring some results from models with atoms to models of $\sf ZF$ (without atoms). These include the existence of a Russell set as well. Other transfer theorems (Pincus, Hall) are equally suitable for the job also.