Since ants, except Fire Ants, are part of Ma Natures recycling machine and play an important role in helping to maintain the environment we live in why do you feel the need to kill them?
If what you have are Fire Ants this web site can aid in controlling them, but they are the only ants that do need control as long as the ants stay out side.

Here is a link that might be useful: About Fire Ants

dinajean, would it be possible for you to capture some of these ants and take them to your county extension office for identification?

As to whether they sting or not, that depends upon the species of ant. All species can bite, but not all of them sting.

I am sure I can capture one or two and bring to extension office, that is indeed a good idea.
They are large like carpenter ants and have made several areas of large holes in my lawn that cave in when I walk over it, I am not trying to be inhumane, I am trying to save my lawn.
That said, what would y'all do?

I'd have them identified so that I could make a smarter decision about what to do. You've already learned that they don't react to fire ant bait. I wouldn't apply another pesticide without knowing 'who' you're up against. Make sense?

Be sure to let us know what you find out.

What are those big red things? Make holes in yard.