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.)
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?