Mirsky's Theorem states that the minimum number of chains needed to cover every element of a finite partially ordered set exactly equals the size of the set's longest antichain, a collection of elements no two of which are comparable to each other. Named for Leon Mirsky, it is the order-theoretic dual of Dilworth's Theorem, which gives the same equality with the roles of chains and antichains reversed.
Facts
StatementFor every finite partially ordered set, the height equals the minimum number of antichains into which the set may be partitioned. 1 Classification
Statement Form Connections
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Mirsky's theorem (Wikipedia)
Sources
1. Mirsky's theorem (Wikipedia)
The theorem
Mirsky's theorem states that, for every finite partially ordered set, the height also equals the minimum number of antichains (subsets in which no pair of elements are ordered) into which the set may be partitioned.
Naming attribution
It is named for Leon Mirsky (1971).
In Branch: Combinatorics, Lead sentence
In mathematics, in the areas of order theory and combinatorics, Mirsky's theorem characterizes the height of any finite partially
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