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(I am assuming choice.)

Suppose that ${\mathbb P}=(P,{\lt})$ is a partially ordered set (a poset), and that $\kappa\le\lambda$ are ordinals. The notation $$ {\mathbb P}\to(\kappa)^1_\lambda $$ means that whenever $f:P\to\lambda$, we can find some $i\in\lambda$ and some subset $H$ of $P$ such that $(H,{\lt})$ is order-isomorphic to $\kappa$ and $f(a)=i$ for all $a\in H$.

Theorem. Suppose that $P$ is a poset and $P\to(\kappa)^1_\kappa$, where $\kappa$ is an infinite cardinal. Then $P\to(\alpha)^1_\kappa$ for all $\alpha\lt\kappa^+$.

This is due to Galvin (unpublished). A nice combinatorial argument is presented in Stevo Todorcevic, "Partition relations for partially ordered sets", Acta Math. 155 (1985), no. 1-2, 1-25.

In fact, Galvin result is that:

For any poset $P$, $$ P\to(\kappa)^1_\lambda $$ implies $$ P\to(\alpha)^1_\lambda $$ whenever $\kappa\le\lambda$ are infinite cardinals and $\alpha\lt\kappa^+$.

This can be proved by collapsing $\lambda$ to $\kappa$ with a $\kappa$-closed forcing, noting that in the extension $P\to(\kappa)^1_\kappa$, so (by the theorem) $P\to(\alpha)^1_\kappa$, and using the closure of the forcing to find such a homogeneous set of type $\alpha$ in the ground model.

My question:

How can we prove Galvin's result without appealing to a forcing argument?


Very briefly, Stevo's proof of the theorem proceeds as follows: Given a poset $P$, let $\sigma'P$ be the collection of injective sequences $\tau$ whose domain is a successor ordinal and whose range is strictly increasing in the ordering of $P$. This is a poset under the "initial segment" ordering of sequences. Stevo proves two results:

  1. If $P\to(\kappa)^1_\kappa$ holds, then $\sigma' P\to(\kappa)^1_\kappa$ holds.
  2. If $\sigma' P\to(\alpha)^1_\gamma$ holds, then $P\to(\alpha)^1_\gamma$ holds.

Item 2 is straightforward, and a more general result holds. Item 1 uses a delicate argument and I do not know of a more general statement. The general version of 2 says that many partition relations that hold for $\sigma' P$ must hold for $P$ as well.

The combination of 1 and 2 is very powerful: It says that to prove partition results for posets $P$ satisfying $P\to(\kappa)^1_\kappa$ it suffices to prove the result for trees, for which the combinatorics tends to be much better understood than for arbitrary posets.

For example, for trees $T$ it is essentially obvious that if $T\to(\kappa)^1_\kappa$, then $T\to(\alpha)^1_\kappa$ for all $\alpha\lt\kappa^+$, and the theorem follows.

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Hi Andres. This sounds interesting. But can you elaborate on how you use the ${<}\kappa$-closure of the forcing $Coll(\kappa, \lambda)$ to get a homogeneous set of order type $\alpha$ in the ground model for arbitrary $\alpha < \kappa^+$? I can see how we can use it to recover a well order of order-type $\kappa$ by defining a descending sequence of conditions $\langle p_{\alpha}| \alpha < \kappa\rangle$ where $p_{\alpha}$ decides element # $\alpha$ in the order, but no more. Also, are you assuming $\kappa$ is regular since you're using that $Coll(\kappa, \lambda)$ is ${<}\kappa$-closed? –  Jason Dec 8 '10 at 7:37
    
@Jason : One can assume $\kappa$ regular (though you are right, I didn't comment on it). Then, given a homogeneous set of type $\alpha\in[\kappa,\kappa^+)$, fix a bijection $\varphi$ between $\kappa$ and $\alpha$, and let $p_\beta$ ($\beta<\kappa$) decide the $\varphi(\beta)$-th element of the set. –  Andres Caicedo Dec 8 '10 at 17:40
    
@Andres: Oh yeah, of course. Thanks. –  Jason Dec 8 '10 at 18:00
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1 Answer

up vote 6 down vote accepted

Assume that $P\rightarrow(\kappa)^1_\lambda$, yet for some $\alpha<\kappa^+$, $P\not\rightarrow(\alpha)^1_\lambda$. Accordingly, there is a decomposition $P=\bigcup_{\xi<\lambda}P_\xi$ such that no $(P_\xi,<)$ does contain a chain of type $\alpha$. For some $\xi<\lambda$ we must have $P_\xi\rightarrow (\kappa)^1_\lambda$ as otherwise all of them would be the union of $\lambda$ subsets, each missing a chain of type $\kappa$, and then, takigk union, the same would hold for $P$. Applying Galvin's original theorem, $P_\xi$ contains a chain of type $\alpha$, contradiction.

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Hi Peter. Thanks! This turned out easier than expected. –  Andres Caicedo Dec 8 '10 at 17:36
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