Which Tyres?

Depends on the rim I guess, some carbon rims don't have such a sharp edge, but guess they need to cover their backs.

Full carbon rims and clincher tyres are-- and I know a lot of people here will scream-- not a good combination. There are just too many fundamental issues.. and why? Tubular tyres make so much more sense. The rims can be made stronger and lighter. The tyres are lighter, more reliable and provide a much better interface to the rim. The art of gluing tyres on carbon rims too has been pretty much well explored and poses not much of a problem as it did for some mechanics 20+ years ago. Why bother with clinchers? They might be fine for touring but then carbon is not (riding on a flat tubular tyre with a carbon rim is generally a very poor idea).. So sure.. some carbon clincher rims will abrade the sidewalls of tyres.. have more limits on max. pressures, some rim horns may crack during tyre installment, some may distort under heat, ride a lot rougher, etc etc. etc. and they cost a lot more (even the noname Chinese jobs).. But I think anyone who has selected clincher+carbon has done so accepting these, another, "side effects".