TITLE: Decomposing posets into countably many chains
QUESTION [8 upvotes]: A conjecture of Galvin's is that the following is possible (which I take to mean that the consistency of the following can be proven relative to the consistency of something like ZFC, or ZFC plus some large cardinal):

A partially ordered set $P$ can be decomposed into countably many chains iff the same is true of every suborder $P_0 \subseteq P$ of size at most $\aleph _1$.  

You can see it stated as Conjecture 3.3 in Todorcevic's "Combinatorial Dichotomies in Set Theory".
I'm interested in this conjecture and determining its consistency strength, but in order to get a feel for it I want to first look at it in a special case in which it's supposed to be (according to Todorcevic, if I understood what he told me correctly) provable from ZFC alone.  I'll actually list three special cases of increasing generality; an answer to the last case would be ideal but I'd be happy to see an answer for the first case.

Galvin's conjecture, restricted to posets $P$ which are the Cartesian product of two linear orders (with the obvious product ordering).
GC restricted to posets which are the Cartesian product of countably many linear orders (countable = finite or denumerable).
GC restricted to posets which are subsets of some Cartesian product of countably many linear orders.

REPLY [5 votes]: I've investigated Galvin's Conjecture quite a bit over the past few years. Here is a preprint of mine on the subject. In there, I show that Galvin's Conjecture restricted to finite-dimensional posets (posets which are subsets of some cartesian product of finitely many linear orderings) is equivalent to Rado's Conjecture (which was extensively studied by Todorcevic). Unfortunately, the ℵ0-dimensonal case has eluded me; perhaps you will have better luck...