We know that different frequencies of light in transparent medium is different. For example red light goes faster than blue light in water. What about vacuum? Why are we sure that all frequencies of electromagnetic wave have the same velocity in a vacuum?
Your transparent medium must be absorbing at some wavelength, otherwise it would not have dispersion (this is obvious from the Kramers-Kronig relations). If you have a real vacuum there is nothing that could absorb - hence there can't be dispersion...
Naghi,
Firstly, there is no experimental evidence for the dispersion of light in a vacuum. Starlight traverses vast gulfs of space - and yet absorption features from the nearer stars are not seen to be altered purely as a function of distance.
If the nature of spacetime was such that one had, built into it, a dispersive quality, then that would have severe implications for Relativity - which mandates that light signals have a single limiting speed - irrespecitve of frequency.
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