Separating Combinatorial Principles Using Nonstandard Models The Stable Version of Ramsey’s Theorem for Pairs Does Not Imply the General Version

he main objective of this research is to study the relative strength of combinatorial principles, in particular, the principles related to Ramsay’s theorem. It turns out that the most interesting ones are those weaker than the Ramsey's theorem for pairs. The strength is measured by hierarchies from either recursion theory or reverse mathematics. Let us recall the precise statement of Ramsey's Theorem: Any function f from n-element subsets of the set of natural numbers to natural number k={0,1,…, k-1} has an infinite homogeneous set H, namely, f is constant on n-element subsets of the set H. One informal reading of the theorem says, if we think of f as a k-coloring of the n-element subsets of natural numbers, then there is an infinite set H, whose n-element subsets of have the same color. It is customary to think of this kind of ̀ `for all f there exists H…’ ’ statements as ̀ `our opponent posed a problem (e.g., coloring