Partition relations for partially ordered sets
Stevo B. Todorčević · Acta Mathematica · 1985
Let P be a partially ordered set.If r<to, then [p]r denotes the set of all sequences (al ..... ar) such that al ..... arEP and al<e... <par.If ~' is an ordinal and if ai, i<y r are order types (isomorphism types of linearly ordered sets), then the symbol P (ai)i< r means that for any partition [P]r=oi<rKi there exists an i<y and a chain A~_P such that tpA=a; and [A]r~K~.The negation of the partition symbol is written with -~ instead of---~.Note that if P is a linearly ordered set, then [p]r and P---~(ai)i~<r have the usual meanings.If cti=ct for all i<y, then we write P-~(a)~ instead of P--~(oti)i<y.r This paper is a study of the partition symbol P (a;);<r for partially ordered sets P such that P--~(~)~ for some infinite cardinal ~.Our main result for the case x--t0 is the following theorem which proves a conjecture of Galvin [10; p. 718].THEOREM 1.Let P be a partially ordered set such that P--~(to)~.Then P--,(a) 2 for all a<to 1 and k