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).

>>11
Timing attacks are possible amongst all consumer grade low latency anonymity networks.
The resources required to mitigate them while still keeping the low latency property are greater than most users can spare, since it quite literally requires you to send a constant stream of traffic even when you have nothing to say.

The other ones (bittorrent attack and email password interception) are also common amongst all protocols that are not fully peer to peer. It an application layer attack that is out of the scope of Tor, since it could have been avoided completely had the website's owners used HTTPS instead of plaintext HTTP.

The only attack on there that actually has relevance to Tor's thread model is the one by the engineering school. Tor wrote a blog post discrediting it, but I'm not familiar with the attack so I won't say anything more on it.

The only reason Tor has such a large "weaknesses" section is that Tor has the most researchers working on it. It says nothing of the actual security of the service except that it's well researched.
Please don't let your ignorance create fear like that