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

Meshalkinʹs theorem states that a class of ordered p-partitions of an n-set has at most max(na1,…,ap) members if for each k the kth 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 Meshalkinʹs theorem and Erdősʹs theorem about r-chain-free set families.