  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.
