  How large an antichain can we find inside a given downset in the lattice of
subsets of [n]? Sperner's theorem asserts that the largest antichain in the
whole lattice has size the binomial coefficient C(n, n/2); what happens for
general downsets?
  Our main results are a Dilworth-type decomposition theorem for downsets, and
a new proof of a result of Engel and Leck that determines the largest possible
antichain size over all downsets of a given size. We also prove some related
results, such as determining the maximum size of an antichain inside the
downset that we conjecture minimizes this quantity among downsets of a given
size.
