# Decomposing a poset into directed subposets

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:

Every poset $P$ with the $\kappa$ chain condition is the union of fewer than $\kappa$-many $\kappa$-directed subposets.

For $\kappa = \omega$ the above holds, it is a theorem due to M. Pouzet. The proof hinges on the following fact:

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.

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:

(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.

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:

(CH) If $P$ has the $\omega _1$-cc, then it is a countable union of $\omega _1$-directed subposets.

My question is simply a reference request: does anyone know where I can find a proof of this statement?

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Thanks.${}{}{}$ – Amit Kumar Gupta Mar 3 '11 at 17:35