oxidation states of halogens

that's easy. think about the structures of oxyanions of the halogens, like chlorine(V) fluoride (chlorine pentafluoride, ClF5), a square pyrimidal molecule.

clearly, with only one singly-occupied 3d orbital, chlorine is unable to form 5 bonds as well as accomodate a lone pair. thus, it undergoes sp3d2 hybridization and yields six new orbitals -- 5 singly-occupied, 1 fully-occupied. the five orbitals enables chlorine to form 5 bonds wit 5 fluorine atoms.

similar for other (non-diatomic) interhalogens.

that's the trick. you need vacant low-lying orbitals for hybridization for more vacancies. chlorine, bromine and iodine can do this, but fluorine can't. low-lying favors promotion of electrons to those orbitals, followed by hybridization.