Unfortunately this isn’t an option, would be cool if it was though! All you can do really is just keep refreshing the page to see any new auctions
You may find an RSS feed useful (I don't know the technicalities of them but they're how podcast services know when someone releases a new podcast and makes it available). You can use a chrome extension (e.g. RSS Feed Reader) which, I think, will check the auction address (below) once you add it and notify you when a new thread is posted. https://empireminecraft.com/forums/auctions.42/index.rss
Hm, an interesting idea! I don't think I would use it, though. I do read all threads in the Empire Creativity Corner, and in Other Gaming, but I like to check those manually, whem I've got time.
Forum like an auction isn't possible, but do pay special attention to the "what's new" list and your news sections. This here: Those options can show an overview of recent activities; either general (posts by everyone) or only activities from people you follow (news feed).
