Complete Anonymous Server

Is it possible to prevent a chat server from knowing who is talking to who?

such that there exists a function negl such that
Pr[A(x,y)=1] < 1/2+negl(x,y)

where Pr is probability and A outputs 1 if the server can correctly guess if x and y are communicating. negl is defined as asymptotically negligible (i.e. negl<1/p(x) where p is some polynomial).

Dining cryptographers probably applies here.
It's a method of broadcasting messages between a group of k clients such that it's impossible to determine which of the k clients originated the message.
Although this system is decentralised, so maybe not.

Sounds like what you want to do is just connect over Tor or something.