In the absence of the Axiom of Choice we study countable families of 2-element sets with no choice functions which either have infinite subfamilies with a choice function or no infinite subfamilies with a choice function.

Keywords: Axiom of Choice; Russell-cardinals; anti Russell-cardinals; prime cardinals; amorphous cardinals; permutation models.

Quaestiones Mathematicae 33(2010), 1–9