  Sperner's bound on the size of an antichain in the lattice P(S) of subsets of
a finite set S has been generalized in three different directions: by Erdos to
subsets of P(S) in which chains contain at most r elements; by Meshalkin to
certain classes of compositions of S; by Griggs, Stahl, and Trotter through
replacing the antichains by certain sets of pairs of disjoint elements of P(S).
We unify Erdos's, Meshalkin's, and Griggs-Stahl-Trotter's inequalities with a
common generalization. We similarly unify their accompanying LYM inequalities.
Our bounds do not in general appear to be the best possible.
