My question involves a child custody case from the State of: Kentucky

I am expecting a child soon, and the father has had a lawyer send me a letter requesting me to grant them permission to prove paternity at the time of the baby's birth via umbilical cord blood testing. It says if I don't respond within 5 days, they will seek the courts to settle it. I have not responded and don't plan on it, because I don't want him to establish paternity, or I want to prolong it as long as possible. I don't want him on the birth certificate, and I don't want him to have any rights. He has a serious drug and alcohol history, along with a pretty serious mental health history. He contacted me about 2 months ago and told me he had relapsed and that he had attempted suicide in October and he was seeking short term disability from work because he was mentally unstable- yet he now wants rights to the baby? It scares me to know someone so unstable could potentially have her. I have no criminal background history, nor do I have any history of drug and alcohol abuse. I have a stable job as a nurse, and a stable home. The father and I only dated for a couple weeks,a nd were not together at the time I found out I was pregnant.

How do I apply for sole custody. With his drug and alcohol issues, plus he was arrested for heroin some years ago, do I have a good chance of having sole custody. This is a very stressful thing to think about the last few weeks of my pregnancy, and can't believe I was bombarded by a letter from an attorney.

Advice would be much appreciated. Thank you