(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This is due to Galvin (unpublished).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:

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