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 $\aleph \_1$. 
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> 
> 


You can see it stated as Conjecture 3.3 in [Todorcevic's "Combinatorial Dichotomies in Set Theory"](http://www.math.toronto.edu/~stevo/dichotomies4.pdf).


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.


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



Decomposing posets into countably many chains
I've investigated Galvin's Conjecture quite a bit over the past few years. Here is a [preprint](http://www-personal.umich.edu/~dorais/docs/galvin.pdf) 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...


