A partially ordered set (also poset) is a set S with a binary relation ≤ (sometimes ⊆ is used) such that:

1. (i) a ≤ a for all a ∈ S (reflexivity),

2. (ii) if a ≤ b and b ≤ a then a ≤ c (transitivity),

3. (iii) if a ≤ b and b ≤ a then a = b (antisymmetry).

If for any a and b in S, either a ≤ b or b ≤ a, then the partial order is called a total order, or a linear order. If a ≤ b and a ≠ b, then we also write a < b. Examples of posets include the integers with the usual order or the subsets of a set, ordered by inclusion. If a subset of S is totally ordered, it is called a chain. An antichain is a set of elements that are pairwise incomparable.

The following theorem is due to R. Dilworth (1950). This proof is due to H. Tverberg (1967).

Theorem 6.1.Let P be a partially ordered finite set. The minimum number m of disjoint chains which together contain all elements of P is equal to the maximum number M of elements in an antichain of P.

Proof: (i) It is trivial that m ≥ M.

(ii) We use induction on |P|. If |P| = 0, there is nothing to prove. Let C be a maximal chain in P. If every antichain in P\C contains at most M – 1 elements, we are done. So assume that {a1,…, aM} is an antichain in P\C.