Carbon-enhanced metal-poor (CEMP) stars comprise a large percentage of stars at the lowest metallicities. The stars in the CEMP-no subcategory do not show any s-process enhancement and therefore cannot easily be explained by transfer of carbon and s-process elements from a binary AGB companion. We have performed radial velocity monitoring of a sample of 22 CEMP-no stars to further study the role binarity plays in this type of CEMP star. We find four new binary CEMP-no stars based on their radial velocity variations, thereby significantly enlarging the population of known binaries to a total of eleven. One of the new binary systems is HE 0107-5240, one of the most iron-poor stars known, supporting the binary transfer model for the origin of the abundance pattern of this star. In our sample we find a difference in binary fraction depending on the absolute carbon abundance, with a binary fraction of $47^{,+15,}_{,-14} %$ for stars with higher absolute carbon abundance and $18^{,+14,}_{,,-9} %$ for stars with lower absolute carbon abundance. This potentially implies a relation between a high carbon abundance and the binarity of a metal-poor star. Although binarity does not equate to mass transfer, there is a possibility that a CEMP-no star in a binary system has been polluted and care has to be taken in the interpretation of their abundance patterns. We furthermore demonstrate the potential of Gaia to discover additional binary candidates.
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