Carbon filaments are odd. They CAN be lit at low temperatures (a very dim, reddish glow) in open air without quickly burning through, unlike tungsten.
This is not that much about the temperature where it starts to react with oxygen or so, but about the thermal stability along the filament (in other words about the tendency to form hot spots vs equalize the temperature along the filament).
With a long and thin filament structure heated by a passing current, the highers power dissipation concentrates on spots with the highest resistance (the current is the same along the whole filament, so the local powerr density then depends on the local voltage drop).
Because carbon has a negative temperature coefficient, the highest resistance will be on the coldest places. That means the heat focuses on the coldest places of the filament and when they warm up, it moves elsewhere (well, if the current does not changes).
With the positive temperature coefficient of the tungsten (and many other metals), the spot with the highest resistance is the hottest one, so the heat tends to concentrate there. That means any non-uniformity in the filament and one spot becomes way hotter and it will be that one, which will then receive most of the power.
So although the carbon filament as a whole is way less stable than the tungsten, its tendency to equalize the temperature along it's length allows you to manually adjust the power so, it glows steadily and yet do not reach the combustion temperature. On the other hand with the tungsten, it is the hot spot formation, which makes it to very quickly exceed the ignition temperature and so causes the filament to burn.
Normally when the thing is intended to operate as an incandescent lamp, majority of the heat dissipation goes via radiation and that goes with the very steep T^4 function, so stabilizes both local temperatures of the tungsten, as well as overall temperature of the carbon. But on free air and way lower temperatures the radiation part is way weaker than the convection, which is governed by way less steep function.
By the way the tendency to hot spot formation of the materials with positive TC and the overall thermal instability of the materials with a negative Tc is the main reason, why the rather low temperature incandescent heaters operating on free air use an alloy (beside it's high temperature robustness on the free air) with very low TC for the heater (= filament) wire.