  The set consensus problem has played an important role in the study of
distributed systems for over two decades. Indeed, the search for lower bounds
and impossibility results for this problem spawned the topological approach to
distributed computing, which has given rise to new techniques in the design and
analysis of protocols. The design of efficient solutions to set consensus has
also proven to be challenging. In the synchronous crash failure model, the
literature contains a sequence of solutions to set consensus, each improving
upon the previous ones.
  This paper presents an unbeatable protocol for nonuniform k-set consensus in
the synchronous crash failure model. This is an efficient protocol whose
decision times cannot be improved upon. Moreover, the description of our
protocol is extremely succinct. Proving unbeatability of this protocol is a
nontrivial challenge. We provide two proofs for its unbeatability: one is a
subtle constructive combinatorial proof, and the other is a topological proof
of a new style. These two proofs provide new insight into the connection
between topological reasoning and combinatorial reasoning about protocols,
which has long been a subject of interest. In particular, our topological proof
reasons in a novel way about subcomplexes of the protocol complex, and sheds
light on an open question posed by Guerraoui and Pochon (2009). Finally, using
the machinery developed in the design of this unbeatable protocol, we propose a
protocol for uniform k-set consensus that beats all known solutions by a large
margin.
