33% Impossibility Result

In general (assuming a partially-synchronous system model), a protocol can achieve safety all the time and additionally liveness in synchronous conditions if and only if $n\ge 3f+1$ (equivalently, $f<\frac{n}{3}$)

This result holds despite the assumption of PKI (unlike the PSL-FLM result), so this bound must be driven by the possibility of unbounded message delays.

Intuition:

1. We can’t wait indefinitely for all nodes to respond (one valid strategy for Byzantine nodes is to never respond, even after GST) so realistically we can only wait to hear from $n-f$ nodes before deciding on the next possible action
2. But we can’t say for certain the $f$ nodes are actually Byzantine (they could be honest nodes that are congested), thus of the $n-f$ nodes, more than half should be honest, $f<\frac{1}{2}(n-f)$ or equivalently $f<\frac{n}{3}$