  Meshalkin's theorem states that a class of ordered p-partitions of an n-set
has at most $\max \binom{n}{a_1,...,a_p}$ members if for each k the k'th parts
form an antichain. We give a new proof of this and the corresponding LYM
inequality due to Hochberg and Hirsch, which is simpler and more general than
previous proofs. It extends to a common generalization of Meshalkin's theorem
and Erdos's theorem about r-chain-free set families.
