Negative Partition Relations for Ordinals ωωα

Abstract

For ordinals :, ;, and #, the expression : Ä % ( ;, #) 2 means there is a partition of the pairs from :, [:] 2 =2 0 _ 2 1 such that for any X :, if the order type of X is ; then [X] 2 3 2 0 and if the order type of X is # then [X] 2 3 2 1 . It is shown that if :<| 1 is multiplicatively decomposable, then | : Ä % (| : , n) 2 for n=4 or n=6, depending on the degree of decomposability of :.

Countable partition ordinals

“…Larson's proof is the canonical proof of the theorem and our results are based the techniques and ideas of her proof. About the same time Galvin and Larson [2] showed that if α is a countable partition ordinal then α is either equal to ω 2 , or α is of the form ω ω β for some countable β. Thus the question becomes: for which countable β does ω ω β → (ω ω β , 3) 2 ?…”

“…Note that (2) says that the bottom nodes are labeled with natural numbers (we use singleton sets for later convenience) which are increasing from left to right.…”

Open problems of Paul Erd�s in graph theory

“…if β is the sum of two indecomposables. Darby [59] proved that A down-up matching in an ordinal graph is a matching of a set A with a set B where every element of A is less than all elements of B, denoted by A < B. Suppose for every graph on an ordinal α there is either an independent set of type β or a down-up matching from a A to a set B.…”