Dilworth's Theorem, states that in a poset the size of the largest antichain is the same size as the smallest chain decomposition. If I am understanding this correctly then, In a totally ordered set, there are no antichains. So the biggest antichain would be of size $0$ and therefore, the smallest chain decomposition would be of size $0$. This seems like a contradictory conclusion to me, so is there some sort of exceptions for posets with no antichains?
2026-04-29 16:15:54.1777479354
Dilworth's Theorem for Totally Ordered Sets
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