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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.
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Amit, do you have any references or know where the conjecture was posed? –  Andres 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
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up vote 5 down vote accepted

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...

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Thanks! You mention in your paper that GC for finite dimensional posets is equivalent to RC, and that a supercompact cardinal gives an upper bound on the consistency strength of RC. Do you know of a lower bound? I think Todorcevic mentioned that RC implies the failure of square everywhere, but what about a lower bound in terms of large cardinals? (related question: mathoverflow.net/questions/53467/…) –  Amit Kumar Gupta Jan 27 '11 at 8:52
    
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
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