I’ve been thinking about this problem ever since an old post of Qiaochu’s first raised the question, and I’ve been frustrated by my inability to solve it. I could post it on MO, but I sort of already have, and anyway it raises questions which are too ill-formed right now to be right for MO. So anyway, here we go:

A lot of problems in extremal combinatorics correspond to finding large antichains in partially ordered sets. (By the way, all posets in this post will be assumed to have a least element.) Classically speaking, Dilworth’s theorem completely characterizes the size of antichains in posets; however, this is often tricky to apply, since it’s not always clear whether a partition into chains is minimal. In addition, it’s sometimes the case (particularly with infinite posets) that there are infinitely long antichains, but a nontrivial bound should still be attainable. The way to get by both of these obstacles is to assign weights to the elements of the poset, and rather than looking for large antichains, we look for antichains with high total weight.

The classical example of this solves the problem of finding the largest antichain in the lattice of subsets of a given finite set — the content of Sperner’s theorem. (more…)