  A super-solar fluorine abundance was observed in the carbon-enhanced
metal-poor (CEMP) star HE 1305+0132 ([F/Fe] = +2.90, [Fe/H] = -2.5). We propose
that this observation can be explained using a binary model that involve mass
transfer from an asymptotic giant branch (AGB) star companion and, based on
this model, we predict F abundances in CEMP stars in general. We discuss wether
F can be used to discriminate between the formation histories of most CEMP
stars: via binary mass transfer or from the ejecta of fast-rotating massive
stars. We compute AGB yields using different stellar evolution and
nucleosynthesis codes to evaluate stellar model uncertainties. We use a simple
dilution model to determine the factor by which the AGB yields should be
diluted to match the abundances observed in HE 1305+0132. We further employ a
binary population synthesis tool to estimate the probability of F-rich CEMP
stars. The abundances observed in HE 1305+0132 can be explained if this star
accreted 3-11% of the mass lost by its former AGB companion. The primary AGB
star should have dredged-up at least 0.2 Msun of material from its He-rich
region into the convective envelope via third dredge-up, which corresponds to
AGB models of Z ~ 0.0001 and mass ~ 2 Msun. Many AGB model uncertainties, such
as the treatment of convective borders and mass loss, require further
investigation. We find that in the binary scenario most CEMP stars should also
be FEMP stars, that is, have [F/Fe] > +1, while fast-rotating massive stars do
not appear to produce fluorine. We conclude that fluorine is a signature of
low-mass AGB pollution in CEMP stars, together with elements associated with
the slow neutron-capture process.
