A partial order on a set X is a reflexive, antisymmetric, and transitive relation R.
A set X on which a partial order <= is defined is sometimes referred to as a
"partially ordered set"(or sometimes simply as a "poset") and denoted by (X, <=)
In a "chain" every of any elements is comparable.
In an "antichain" every pair of elements is incomparable.
Let (X, <=) be a finite partially ordered set, and let r be the largest size of a chain.
Then X can be partitioned into r but no fewer antichains.
The dual theorem is generally known as Dilworth's Theorem.
Let (X, <=) be a finite partially ordered set, and let m be the largest size of an antichain.
Then X can be partitioned into m but no fewer chains.
