The problem is what's the minimum number of total order subsets can a partial order set partition into?
For example, (1,2) and (3,4) are comparable i.e. (1,2) < (3,4), and (1,2) and (2,1) are incomparable. For the partial order set { (1,2), (3,4), (2,1)} we can partition into two subsets {(1,2), (3,4)} and {(2,1)}, which both are total order sets.
I searched on the web and found little related documents. The question is that whether any works or solutions on this problem exist?
Given a finite partially ordered set, Dilworth's theorem says that the minimal number of totally ordered subsets needed to cover the entire set is equal to the size of the largest antichain (also known as the width of the set). An antichain is a subset where no element is comparable to any of the other elements in the antichain.