It appears right after I cut the grass. I do not really see any ants. The holes are the size of 3mm or so.

We have small holes in the lawn, 1" around and 6" deep, no critter is present, the holse are starting to get bothersome. Any ideas what they are?

Someone dug up a tree or shrub and filled the hole with compost or mulch. While it seemed like a good idea at the time, when compost or mulch decomposes, it has no volume left. In order to fill that hole you will need a mineral based soil. Sand and clay are minerals. Your dirt is probably minerals. Eventually you will have it filled. I'd continue doing what you're doing until it stops. Or you could dig out the 18 inches and refill all at once. If you do that be sure to mound the top of it because even minerals will continue to settle for three years.

Yes, at least the people in Alabama and Philadelphia can eliminate one cause of holes dug in their lawns; bandicoots. For want of a detailed description, bandicoots are like little kangaroos that scratch around at night looking for grubs, roots, etc. When you sit out at night they will hop silently around your feet but if you make a sudden move they will spring away at lightning speed. Back to the holes, I have seen similar sized holes, about 25mm (an inch for those who haven't caught up with metrics) (cheeky!!) and they are usually spider holes, but can also be dug by wasps, beetles and probably a lot of other insects. Whatever they are, I hope you leave them to "do their thing" Petelong.