oxygen

Basically, because oxygen (O2) can donate electrons to the other molocules in the combustion process (REDOX reaction). Molecular O2 splits into two atomic O's and each of those has two electrons it can share with the carbon or nitrogen atoms to form more stable molocules like CO2, NO2, etc.

By definition? If H2 and F are combined, its a REDOX reation that gives off heat, is it combustion?

H2 plus F2 is not considered combustion as the usual definition for that is the rapid oxidation of a compound with O2 but it is REDOX. The ability to give off heat or to absorb heat from the surrounding environment (exothermic and endothermic) is a different property of the reactions.

REDOX is focused on oxidation numbers and most often with electron transfers.

WIKI.com has good definition of REDOX but a more elusive one for combustion.

now you see why I said "by definition" we normal call a redox with just O2 combustion.

here's one, what do you call this reaction:

2(C2H4) = 3C + CH4 + 2H2 + heat (the pressure tripples too)