Frankliin wrote: << Here's a related problem. Can you describe a *maximal* uncountable antichain? More difficult: describe an (uncountable) maximal antichain where every set is infinite. (I don't know the answer to this second question. It might be: no, no such description is possible.)

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This lasts kind of antichain can be shown to exist using Zorn's lemma, but here's an construction of an uncountable maximal antichain using only infinite subsets of Z: It consists of a) all subsets intersecting every set {2n, 2n+1} in exactly one element and b) all subsets of the form S_n = Z - {2n, 2n+1}. This is clearly an antichain. If any other subset X could be added, X would contain at least one element from each {2n, 2n+1} to avoid being a subset of one of the S_n's. But this means X has a subset of type a), contradiction. --Dan P.S. Here's an old cardinality puzzle in the same vein: What's the largest size of a collection of subsets of Z such that any two of them intersect in a finite set?