Red eye is caused by the light from your flash reflecting off the blood vessel rich rear inner surface of the eye and back into the camera lens. The basic rule of physics is: the angle of incidence = the angle of reflection.
Two factors affect red eye. The opening size of the iris and the distance between the flash and the lens. The further away the flash is from the lens the better in regard to red eye. When the flash is very close to the lens as it is with todays very compact cameras, the angle in which the light reflects back forms a very small cone allowing the light to reflect off the back of the eye and nearly straight back into the camera lens. When the flash is further away the angle of reflection is greater and the light reflects at an angle that does not reach the lens.
If the Iris opening is small, it is less likely that the light will be able to reflect back out of the eye and into the lens.
I've attached an illustration but I don't know if it will upload. The blue lines represent a close lense/flash combination. The green lines show what would happen if the flash were away from the lense. To make matters worse, if you shoot with your camera's lens zoomed in a telephoto mode while using the flash you narrow that cone even more.
Try to turn on any lighting you can to reduce the subject's iris size, don't use flash together with zooming your telephoto lens and you can try using the red eye reduction mode in the camera. It pre-flashes the flash to try to close down the subject's iris. I've never seen it work very well but it's worth giving a try.
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