Partially Ordered Sets · Dilworth's Theorem

Lesson 3

Nikolai Chukhin · Alexander S. Kulikov

Both the maximum chain size and the minimum size of an antichain covering correspond to the height of a poset. For our previous toy example, the maximum size of a chain is four: \(\{2, 4, 8, 0\}\).

The whole set can also be covered with the following four antichains: \[\{2, 3, 5, 7\}, \quad \{4\}, \quad \{8, 6, 9\}, \quad \{0\}.\]

For the following poset, these two quantities (the maximum size of a chain and the minimum size of an antichain covering) are also equal.