The RHS is the International Cultivar Registration Authority for Dianthus, including all types of carnations and pinks
A list of cultivar names was published in The International Dianthus Register, and it is kept up-to-date with regular supplements. As well as names, these publications include wherever possible a brief description, parentage, date of raising and classification for each cultivar.
Download registration forms here:
Application form to register a Dianthus cultivar name
The RHS publishes a list of cultivar names, most recently in The International Dianthus Register 2016 (6.1MB pdf).
The first supplement (312kB pdf) was published in 2018.
The RHS published a list of cultivar names in The International Dianthus Register 1982 (now out of print) and subsequent supplements, some of which are available as downloads from the link below.
International Dianthus Register (1982) - supplements