An Intermission: Ramsey’s Theorem

Abstract

Some notation, special to the present discussion, ought to be introduced. If A and B are subsets of the set N of natural numbers, then we write A < B whenever a < b holds for each a ∈ A and b ∈ B. The collection of finite subsets of A is denoted by ℘ _<∞(A) and the collection of infinite subsets of A by ℘ _∞(A). More generally for A,B ⊆ N we denote by ℘ _<∞(A,B the colelction { X ∈ ℘ _<∞ (N) : A ⊆ X A ∪ B, A < X\ A } and by ℘ _∞ (A,B) the collection { X ∈ ℘ _∞ (N): A ⊆ X ⊆ A ∪ B, A < X\ A }.