Partial choice functions for families of finite sets
  Let m>2 be an integer. We show that ZF + "For every integer n, Every countable family of non-empty sets of cardinality at most n has an infinite partial choice function" is not strong enough to prove that every countable set of m-element sets has a choice function. In the case where m=p is prime, to obtain the independence result we make use of a permutation model in which the set of atoms has the structure of a vector space over the field of p elements. When m is non-prime, a suitable permutation model is built from the models used in the prime cases. 
Eric J. Hall, Saharon Shelah