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They allow you to just start a new subreddit. Or download the source and run your own....

Already exists (though a bit dead) http://www.reddit.com/r/slashdot

the reddit stack is open source (beyond their secret sauce anti-spam stuff) so you can easily run run your own reddit clone already.

It does. You can start your own subreddit, point your own domain at it, and then style it to your liking.

It's not super obvious, but the functionality has actually been there for years.

> similar to how you can pay for a white label Stackoverflow subdomain

[citation needed]

You cannot pay for a white label subdomain. You can however license the engine for internal use only (but pricing makes it only make sense for large institutions).

citation: http://meta.stackoverflow.com/questions/16054/is-the-stack-e...

It looks like at one point, they did used to offer whitelabels:

http://meta.stackoverflow.com/questions/83591/how-to-run-a-p...

Sorry I couldn't find a better link on short notice :)