Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

You can look at: S. Todorcevic, Directed sets and cofinal types.

share|improve this answer
    
Thanks.${}{}{}$ –  Amit Kumar Gupta Mar 3 '11 at 17:35
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.