Kanamori–McAloon theorem

In mathematical logic, the Kanamori–McAloon theorem, due to, gives an example of an incompleteness in Peano arithmetic, similar to that of the Paris–Harrington theorem. They showed that a certain finitistic theorem in Ramsey theory is not provable in Peano arithmetic (PA).

Statement
Given a set $$ s\subseteq\mathbb{N} $$ of non-negative integers, let $$\min(s)$$ denote the minimum element of $$s$$. Let $$[X]^n$$ denote the set of all n-element subsets of $$X$$.

A function $$f:[X]^n\rightarrow\mathbb{N}$$ where $$X\subseteq\mathbb{N}$$ is said to be regressive if $$f(s)<\min(s)$$ for all $$s$$ not containing 0.

The Kanamori–McAloon theorem states that the following proposition, denoted by $$(*)$$ in the original reference, is not provable in PA:


 * For every $$n,k\in\mathbb{N}$$, there exists an $$m\in\mathbb{N}$$ such that for all regressive $$f:[\{0,1,\ldots,m-1\}]^n\rightarrow\mathbb{N}$$, there exists a set $$H\in[\{0,1,\ldots,m-1\}]^k$$ such that for all $$s,t\in[H]^n$$ with $$\min(s)=\min(t)$$, we have $$f(s)=f(t)$$.