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.
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