But that doesn’t mean there are no three-colored cats. Just the likelihood of their appearance is significantly small, roughly like red cats.

Nearly not even so. Homozigotism by the gene located in the X-chromosome, namely the gene of red color, is very common than not just trisomy on the X-chromosome, but also accompanied by heterozigotism by the gene of color in the double X-chromosome in the male. Considering that for the female red color - a variant of the norm, and for the male three-color color indicates the presence of a serious deviation, manifested in particular by infertility, - you can easily produce a breed, all representatives of which will be red regardless of gender, but you will not be able to produce a breed of three-color cats.