Partial choice functions for families of finite sets

Let p be a prime. We show that ZF + “Every countable set of p-element sets has an infinite partial choice function” is not strong enough to prove that every countable set of p-element sets has a choice function, answering an open question from [1]. The independence result is obtained by way of a permutation (FraenkelMostowski) model in which the set of atoms has the structure of a vector space over the field of p elements, and then the use of atoms is eliminated by citing an embedding theorem of Pincus. By way of comparison, some simpler permutation models are considered in which some countable families of p-element sets fail to have infinite partial choice functions.