Your pup is 4 years old and has only had the rash for a year, right? Is there anything different in this last year? Did you change environments, food, lawn treatments, spray house for insects, start having him wear clothing, change shampoo or other hair product? Is he sleeping on a new bed, your bed with different detergents?
I agree with changing to a hypoallergenic food but if the food never changed.... then it would be odd to develop an allergy at 3 years old, when he had been fine on the food all along.
Is there a possibility of fleas? They are often allergic to the fleas and can break out from that. Long after the flea is gone, the pup has to fight this rash. This explains how the allergy is to the proteins in flea saliva
Flea Allergies | Dog Itching | Dog Rashes | Dog Dandruff. It is not dogs with noticeable flea problems that wind up with these rashes but instead it is the dog that rarely comes into contact with them. The dog might be on treatment so fleas do not stay on him at all but will occasionally get on him long enough to bite a few times and get that saliva allergy going.
Other problems that can create a rash like you describe include an allergy/reaction to flea prevention product, mites, fungus infections. Since it is going away when on antibiotics and coming back when the antibiotics are done, there is obviously infection, but that can accompany a rash from any source. It could also be Folliculitis -- there is a picture here:
Skin Problems in Dogs Slideshow: Mange, Allergic Dermatitis, Impetigo, Ringworm, More 4/19.
I know you are concerned about repeated bouts of antibiotics, have you considered topical antibiotics? That way, it would not be a systemic treatment but rather to the rash itself. I would def. ask the vet about trying this for an extended time. Seems like maybe previous antibiotic treatments have not been strong enough or long enough to eradicate the problem (since you said it comes back as soon as you stop the antibiotics).
If this vet will not help find the source of this rash, you might have to find another or ask for a referral to a canine dermatologist as another suggested Over a year -- you should at least have a cause by now!