linguistic:voice_change
Height and width

A maximal antichain is an antichain that is not a proper subset of any other antichain. A maximum antichain is an antichain that has cardinality at least as large as every other antichain. The width of a partially ordered set is the cardinality of a maximum antichain. Any antichain can intersect any chain in at most one element, so, if we can partition the elements of an order into k {\displaystyle k} chains then the width of the order must be at most k {\displaystyle k} (if the antichain has more than k {\displaystyle k} elements, by the pigeonhole principle, there would be 2 of its elements belonging to the same chain, a contradiction). Dilworth's theorem states that this bound can always be reached: there always exists an antichain, and a partition of the elements into chains, such that the number of chains equals the number of elements in the antichain, which must therefore also equal the width. Similarly, one can define the height of a partial order to be the maximum cardinality of a chain. Mirsky's theorem states that in any partial order of finite height, the height equals the smallest number of antichains into which the order may be partitioned.