Partial choice functions for families of finite sets

Let m >= 2 be an integer. We show that ZF + "Every countable set of m-element sets has an infinite partial choice function" is not strong enough to prove that every countable set of m-element sets has a choice function, answering an open question from [D-R]. (Actually a slightly stronger result is obtained.) The independence result in the case where m = p is prime is obtained by way of a permutation (Fraenkel-Mostowski) model of ZFA, in which the set of atoms (urelements) has the structure of a vector space over the finite field F-p. The use of atoms is then eliminated by citing an embedding theorem of Pincus. In the case where m is not prime, suitable permutation models are built from the models used in the prime cases.