Red tattoo inks are the most likely to elicit an allergic reaction from your clients. Some contain iron oxide - also known as common rust - which creates a rusty red colour, and is also found in brown and orange pigments. They might also contain nickel, which can be a potential allergy risk for those sensitive to the metal.
For a more vibrant red, cadmium and cinnabar are used. In some cases, cadmium may cause redness, itching and flaking, but is unlikely to be more than a discomfort.
If your client is concerned about being allergic to red tattoo ink, you can give them a patch test 24 hours before the appointment to see how their skin reacts.