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A Result in Dual Ramsey Theory

✍ Scribed by Lorenz Halbeisen; Pierre Matet

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
86 KB
Volume
100
Category
Article
ISSN
0097-3165

No coin nor oath required. For personal study only.

✦ Synopsis

We start by introducing some notation. We conform to the usual practice of identifying the least infinite ordinal o with the set of non-negative integers.

Given a; b4o; a partition of a into b blocks is an onto function X : a ! b such that minðX À1 ðfngÞÞominðX À1 ðfmgÞÞ whenever nomob: Thus, the blocks of X are ordered as their leaders (i.e., their least elements).

The leader function ' : ðaÞ b  b ! a is defined by 'ðX ; mÞ :¼ minðX À1 ðfmgÞÞ: Hence, the function m/'ðX ; mÞ enumerates the leaders of X in increasing order.

Given X 2 ðaÞ b and Y 2 ðaÞ g ; where a; b; g4o; we let Y 4X if Y is coarser than X ; i.e., each block of Y is a union of blocks of X :

Given a; b; g4o and X 2 ðaÞ b ; ðX Þ g :¼ fY 2 ðaÞ g : Y 4X g: Given a; b4o and koo; ðaÞ b k denotes the set of all X 2 ðaÞ b such that (a) X À1 ðfngÞ is finite if k4nob; and (b) maxðX À1 ðfngÞÞo'ðX ; n þ 1Þ if k4n and n þ 1ob: 394

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