A Shorter, Simpler, Stronger Proof of the Meshalkin-Hochberg-Hirsch Bounds on Componentwise Antichains

Meshalkin’s theorem states that a class of ordered p-partitions of an n-set has at most max ( n a1,...,ap ) members if for each k the k 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 Erdős’s theorem about r-chain-free set families.