I'm confused because you don't treat a cold with antibiotics. A cold is not bacterial; it's a viral infection, and antibiotics are useless.
So is your doctor just pushing antibiotics at what he or she agrees is just a cold? If so, get a new doctor who has a brain and isn't trying to just placate you.
If your doctor is giving your son antibiotics because this is a real bacterial infection that will respond to antibiotics -- find out what the infection really is, because the doctor should not be referring to it as a cold.
I agree with others that this could be allergies but again -- you do not treat allergies with antibiotics. Useless. You could get your son tested just to rule out allergies, and the tests are not always all that bad --my daughter had a simple blood test for allergies. Your son could even have allergies to something AT this day care; is it a home day care where there could be pets and pet dander? Even if pets are kept away from the day care areas, the presence of dander can set off those who are sensitive. Carpeting that has mold beneath? Mold in the walls? Most commercial facilities will be OK but if it's a home day care, there could be other issues going on like pets or mold in the house. Most likely the issue really is an infection that the kids are passing around but if it never ever ends, the day care itself could be an issue.
But mostly I'm wondering exactly what was diagnosed, and whether your doctor is just shoving antibiotics at a cold --? Or did those not start until the doctor said it was bacterial?