Crash Omission Byzantine
Synchrony $f<n$ is possible
$f+1$ round executions must exist
$f \geq n/2$ is impossible $f<n/2$ possible with PKI / PoW
$f \geq n/3$ impossible without PKI/PoW
Partial Synchrony $f \geq n/2$ is impossible $f<n/2$ is possible $f<n/3$ is possible
$f \geq n/3$ is impossible
Asynchrony non terminating executions must exist $f<n/2$ possible in $O(1)$ expected $f<n/3$ possible in $O(1)$ expected

Here $n$ is the number of parties, and $f$ is the number of parties that the adversarial threshold. Recall that Synchrony $\subseteq$ Partial Synchrony $\subseteq$ Asynchrony. Similarly that Crash $\subseteq$ Omission $\subseteq$ Byzantine. Therefore,

  1. Any upper bound holds if we go down and/or to the left. e.g., the $O(1)$ expected round upper bounds under asynchrony also hold in partial synchrony and synchrony.
  2. Any lower bound holds if we go up and/or to the right. e.g., the impossibility of $f \geq n/3$ carries with Byzantine adversaries in partial synchrony carries over to asynchrony and the $t+1$ round lower bound carries over from crash to omission and Byzantine.

Acknowledgments: many thanks to Kartik Nayak for help with this post!

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