Related Work. We are aware of three other proofs of the k-set agreementlowerbound. Chaudhuri, Herlihy, Lynch, and Xxxxxx [7] gave the ﬁrst proof. Their proof consisted of taking the standard similaritychain argument used to prove the consensus synchronous lower bound and running that argument in k dimensions at once to construct a subset of the reachablecomplex to which a standard topological tool called Sperner’s Lemma can be applied to obtain the desired impossibility. While their intuition is geometrically compelling, it required quite a bit of technicalmachinery to nail down the details. Herlihy, Rajsbaum, and Xxxxxx [15] gave a proof closer to our “round-by- round” approach. In fact, the round operator that we deﬁne here is exactly the round operator they deﬁned. Their connectivityproof for the reachable complex was not easy, however, and the inductive nature of the proof did not reﬂect the iterative nature of how the reachable complex is constructed by repeatedly ap- plying the round operator locally to a global state S. The notion of an absorbing poset used in this paper dramatically simpliﬁes the connectivity proof. Gafni [12] gave another proof in an entirely diﬀerent style. His proof is based onsimplereductions between models, showing that the asynchronous model can simulate the ﬁrst few rounds of the synchronous model, and thus showing that the synchronous lower bound follows from the known asynchronous impossibility result for set agreement [4,16,21]. While his notion of reduction is elegant, his proof depends on the asynchronous impossibility result, and that result is not easy to prove. We are interested in a simple, self-contained proof that gives as much insight as possible into the topological behavior of the synchronous model of computation. Round-by-round proofs that show how the 1-dimensional (graph) connec- tivity evolves in the synchronous model have been described by Aguilera and Xxxxx [1] and Moses and Xxxxxxxx [18] (the latter do it in a more general way that applies to various other asynchronous models as well) to prove consensus impossibility results. These show how to do an elegant FLP style of argument, as opposed to the more involvedbackward inductive argument of the standard proofs [10,8,9]. They present a (graph) connectivity proof of the successors of a global state. Thus, our proofs are similar to this strategy in the particularcase of k = 1, but give additionalinsights because they show more general ways of organizing these ...