Greene's theorem (http://ac.els-cdn.com/0097316576900789/1-s2.0-0097316576900789-main.pdf?_tid=3b618666-d514-11e5-b014-00000aacb362&acdnat=1455672037_585aef34d8038d78798742ec5858d8f6) and the Greene-Kleitman theorem (http://ac.els-cdn.com/0097316576900777/1-s2.0-0097316576900777-main.pdf?_tid=69f23e94-d514-11e5-8ffd-00000aacb35f&acdnat=1455672115_9d28b3c1302190aec361c8aad6f7da12) are remarkably deep theorems that hold for any finite partially ordered set.


**Greene's theorem.** Let $P$ be an $n$-element poset. Let
$\lambda_1+\cdots+\lambda_k$ be the largest size of a union of $k$
chains of $P$. Let $\mu_1+\cdots+\mu_k$ be the largest size of a union
of $k$ antichains. Let $\lambda=(\lambda_1,\lambda_2,\dots)$ and
$\mu=(\mu_1,\mu_2,\dots)$. Then $\lambda$ and $\mu$ are conjugate partitions, i.e., they are weakly decreasing, and the Young diagram of 
$\mu$ is the transpose of that of $\lambda$.

To see the subtlety of this result, there is for instance a
nine-element poset with $\lambda=(5,3,1)$, but $P$ is not a union of a
5-element chain and a 3-element chain.

The fact that $\mu_1$ is the number of parts of $\lambda$ is
*Dilworth's theorem*: the size of the largest antichain of $P$ is
equal to the least number of chains whose union is $P$.