Failure patterns in modern parallel and distributed systems are not necessarily uniform. The notion of an adversary scheduler is a natural way to extend the classical wait-free and t-faulty models of computation. A well-established way to characterize an adversary is by its set of cores, where a core is any minimal set of processes that cannot all fail in any execution. We show that the protocol complex associated with an adversary is (c - 2)-connected, where c is the size of the adversary's smallest core. This implies, among other results, that such an adversary can solve c-set agreement, but not (c - l)-set agreement. The proofs are combinatorial, relying on a novel application of the Nerve Theorem of modern combinatorial topology.
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