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...



Chains intersecting antichains in finite posets
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.


