Those assholes just won't stop undercutting you! You don't know what to do! Well I do. Become a detective and figure out their toons, that is what.
The first thing you should do (as most experience AH players will tell you) is to put your competitor on your friends list.
Check out this character's guild. Are there several other characters? Sweet. Do a /who and see who is on in the guild besides your competitor. Refresh it right when your competitor logs off. Then refresh it again for a minute or so. Is there a new character in the search? That is probably your competitor! Put his ass on your friend list too. As a side note, there is an addon that makes your friend list universal across your characters. This is very much needed for spying on the competiton.
You competitor is a bank toon in a guild that has no other members? No problem.
When your competitor logs off, look for characters logging on around mail boxes and possibly using the profession that you are competing for. Character logs on and starts making gems when your competitor logs off? Sweet. That may be your competitor. Put them on your friend's list and monitor them. If you notice the competitor logging off and the other one logging on right afterwards, you have found your person. On the friends list they go. Now that you have an "alt" (more than likely a main) on your friends list, use the /who trick on the guild they are in when your competitor logs off. Any new names pop up? BAM, caught that mofo again.
With a little bit of research, you can be one step ahead of your competition. They will be monitoring your posting character from another of their characters, however you will be monitoring all of theirs. Repost your auctions when they go offline. Or just repost anyways and make them work harder for every gold they make.