How are you assuming flying cars would work? Because I picture flows of "air traffic" along prescribed "air roads", not arbitrary free three-dimensional movement. Which, obviously, allows for (at the very least) rear-ending.
To oversimplify hugely, if you look at commercial aviation, that's near enough how it works. But it's all controlled by ATC who give you a height, direction and maintain separation. A flight plan must have been filed before the flight so ATC know your route. So take all that huge sky and force the planes down little corridors :)
Flying cars are much more likely to be VFR flights in uncontrolled airspace. Go where you like, tell no one, don't hit anyone, stay out of controlled airspace and in sight of ground, responsibility rests with pilot.
I'd be very surprised to see enough flying cars around to reach high enough density to raise risk noticeably outside already high traffic areas.
How are you assuming flying cars would work? Because I picture flows of "air traffic" along prescribed "air roads", not arbitrary free three-dimensional movement. Which, obviously, allows for (at the very least) rear-ending.