### Partial choice functions for families of finite sets

by Hall and Shelah. [HalSh:934]

Fundamenta Math, 2013

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
(Fraenkel-Mostowski) model in which the set of atoms
has the structure of a vector space over the field of p
elements. 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.

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