Your employer is wrong. He is right that he must withhold KY local (city or county) tax. It is not mandatory to withhold KY state tax on an employee who is a full year resident of a reciprocal state (OH).
Give your employer form 42A809 (<a rel="nofollow" target="_blank" href="
https://revenue.ky.gov/Forms/42A809.pdf">https://revenue.ky.gov/Forms/42A809.pdf</a>)
