*You mean 3f +1 total nodes to tolerate f = [(n/3)-1] faulty nodes?

I have looked a bit into the things you pointed out a few days ago. The dPOS in Steem (other than in Bitshares) is not a pure dPOS but a dPOS+BFT or dPOS-BFT or "pipelined dPOS" how it is called in Eos, which makes it actually a quorum. Even though there is no obvious direct each-to-each communication, there is still each-to-each communication.

In dPOS-BFT there is a leap of faith (Vertrauensvorsprung) like in Bitcoin, which makes it a retrospective protocol. After blocks are added (Now comes the quorum) --> the other witnesses have to verify the last irreversible Block (LIB) with at least 2/3. The consensus rule is: that no honest node will join a chain which is not build on the last LIB. Aka. the longest chain rule.

People confuse the consensus mechanism (dPOS) with the oracle (PoS). Consensus is never made by the Oracle. Hash-Cash aka. Proof-of-Work is just an oracle - in fact a special form of proof-of-stake. The consensus is done by the set of consensus rules (fork-choice/longest chain/longest and heaviest chain)

This means it is a quorum by the longest chain rule! Now I understand why Vitalik pointed óut that dPOS-BFT other than dPOS (and I guess dBFT like in Neo) is subject to message complexity!

"There’s a large class of protocols, including traditional BFT algos, where the overhead to each node is proportional to the number of participants. In this proposed DPOS BFT too, it’s the case that “everyone must talk to everyone twice”: with N validators, you get finality after 2N blocks, and each of these blocks needs to be broadcasted to all N participants, so that’s 2N² bilateral communications total." (Vitalik Buterin 2018)

The BFT 2/3+1 safety assumption comes with the use of the round-robin scheduling. When all active nodes add one block per round -> every 3 seconds (the quorum is synchronized by the fixed consortium), THEN a round is k times 3 seconds and the chain grows with --> 3 seconds per block.

Any malicious set of nodes up to 1/3 can create a minority-fork but this malicious fork will only grow as fast as 9 seconds per block (note that the consortium is still one and of size k), ergo the honest chain will still grow with 6 seconds per block. Hence --> 6 seconds is the minimum honesty-decidebility-criterion.

When a malicious minority is bigger than 1/3, we have the equivalent of a 51%-attack vector (not exact the same since bitcoin is binary and this state is a undecidable "pending" state). Because f =1/3 +1 means that there is no honest 2/3 and hence no 6 seconds. So the correctness is violated (even though the system still makes progress in this pending state)