#unwed mother

In at least one State, when an unwed mother seeks government assistance, she is required to name the father. The man can then either admit to being the father, or submit to a DNA test to prove or disprove paternity. The proven father is then responsible to the county for child support  etc. In some cases the woman rattles off a list of potential suspects. My question is "What happens if no father is proven?  Does she still get assistance? ", and "Who pays?"