Greene's theorem 

>Curtis Greene, _Some partitions associated with a partially ordered set_,
Journal of Combinatorial Theory Series A, Vol.20(1) (1976) pp 69–79, doi:[10.1016/0097-3165(76)90078-9](http://dx.doi.org/10.1016/0097-3165%2876%2990078-9)

and the Greene-Kleitman theorem 

>Curtis Greene, Daniel J Kleitman, _The structure of sperner $k$-families_,
Journal of Combinatorial Theory Series A, Vol.20(1) (1976) pp 41–68, doi:[10.1016/0097-3165(76)90077-7](http://dx.doi.org/10.1016/0097-3165%2876%2990077-7)

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 $P$.