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Dilworth's Theorem on Posets states that if $P$ is a poset and $w(P)$ is the maximum cardinality of antichains in $P$ then there exist a decomposition of P of size $w(P)$.

The question is, why this theorem is not trivial?

Consider that there is a whole paper on Annals of Mathematics devoted to it: Dilworth, Robert P. (1950), "A Decomposition Theorem for Partially Ordered Sets", Annals of Mathematics 51 (1): 161–166, doi:10.2307/1969503.

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    If that were Dilworth’s theorem, it *would* be trivial, but in fact Dilworth’s theorem says that the maximum size of an antichain in $P$ equals the minimum size of a partition of $P$ **into chains**. It’s that last requirement that makes the result non-trivial.2011-09-14
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    @Brian, perhaps you'd like to make that comment an answer.2011-09-14

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