Let us say that a poset $P$ is $\mathbf{\kappa}$**-directed** iff every collection of fewer than $\kappa$-many elements in $P$ has an upper bound in $P$. $P$ has the $\mathbf{\kappa}$ **chain condition** iff every antichain in $P$ has size less than $\kappa$. Consider statements of the form:



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> Every poset $P$ with the $\kappa$ chain condition is the union of fewer than $\kappa$-many $\kappa$-directed subposets.
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For $\kappa = \omega$ the above holds, it is a theorem due to **M. Pouzet**. The proof hinges on the following fact:



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> If $P$ is well-founded and has the ccc, then the partial order of downwards-closed subsets of $P$, ordered by inclusion, is well-founded.
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This fact follows from a simple application of Ramsey's theorem. The following statement is not amenable to the same Ramsey-theoretic argument, and is in fact false:



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> (False) If $P$ is well-founded and has the $\omega \_1$-cc, then the partial order of downwards-closed subsets of $P$, ordered by inclusion, is well-founded.
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Therefore, if we replace $\kappa = \omega$ with $\kappa = \omega\_1$ in Pouzet's theorem, we can't apply the same proof. However I've been told (by Todorcevic) that the following is in fact true:



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> (CH) If $P$ has the $\omega \_1$-cc, then it is a countable union of $\omega \_1$-directed subposets. 
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**My question** is simply a reference request: does anyone know where I can find a proof of this statement?



Decomposing a poset into directed subposets
You can look at: S. Todorcevic, Directed sets and cofinal types.


