# Decomposing posets into countably many chains

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.

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.
• Amit, do you have any references or know where the conjecture was posed? – Andrés E. Caicedo Jan 25 '11 at 6:02
• Yup, one of Todorcevic's papers, I've included a link above now. – Amit Kumar Gupta Jan 25 '11 at 6:16

• I haven't looked at RC lower bounds for a while. Since RC implies CC, one lower bound is that of an $\omega_1$-Erdos cardinal, but this is surely very crude. – François G. Dorais Jan 27 '11 at 10:21