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In this paper, we show that the protocol complex of a Byzantine synchronoussystem can remain $(k - 1)$-connected for up to $\lceil t/k \rceil$ rounds,where $t$ is the maximum number of Byzantine processes, and $t \ge k \ge 1$.This topological property implies that $\lceil t/k \rceil + 1$ rounds arenecessary to solve $k$-set agreement in Byzantine synchronous systems, comparedto $\lfloor t/k \rfloor + 1$ rounds in synchronous crash-failure systems. Wealso show that our connectivity bound is tight as we indicate solutions toByzantine $k$-set agreement in exactly $\lceil t/k \rceil + 1$ synchronousrounds, at least when $n$ is suitably large compared to $t$. In conclusion, wesee how Byzantine failures can potentially require one extra round to solve$k$-set agreement, and, for $n$ suitably large compared to $t$, at most that.

Tight Bounds for Connectivity and Set Agreement in Byzantine Synchronous Systems