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I feel a little embarrassed to be asking this question here, since I think it should be much easier than I'm making it, but here goes:

Given a finite poset P, does there necessarily exist some chain that intersects every maximal antichain? (Note: By maximal antichain, I mean that there's no antichain strictly containing our antichain.) The answer seems to be "no" for infinite posets, but I can't find either a reference or a proof when it comes to finite posets.

Sorry if this is an undergrad-homework-level problem...

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On the other hand, there does exists a chain that intersects every antichain of maximum size. –  Richard Stanley Feb 5 '11 at 4:00

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up vote 5 down vote accepted

No.

Consider the poset of subsets of {x,y,z} under inclusion. The maximal chain Ø, {x}, {x,y}, {x, y, z} does not intersect every maximal antichain: it misses the maximal antichain {y}, {x,z}. By symmetry every other maximal chain also misses some maximal antichain.

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Oh, duh! I think I actually had this counterexample myself the other day, but didn't realize that it was important. Hmm, okay... I need to think about this some more, then. Thanks! –  Harrison Brown Oct 20 '09 at 6:41

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