Combinatorial Unprovability Proofs and Their Model-Theoretic Counterparts

Abstract

For a function $f$ with domain $[X{]}^n$, where $X\subseteq \mathbb{N}$, we say that $H\subseteq X$ is canonical for $f$ if there is a $\upsilon \subseteq n$ such that for any $x_0,\dots ,x_{n-1}$ and $y_0,\dots ,y_{n-1}$ in $H$, $f(x_0,\dots ,x_{n-1})=f(y_0,\dots ,y_{n-1})$ iff $x_i=y_i$ for all $i\in \upsilon$. The canonical Ramsey theorem is the statement that for any $n\in \mathbb{N}$, if $f:[\mathbb{N}{]}^n\to \mathbb{N}$, then there is an infinite $H\subseteq \mathbb{N}$ canonical for $f$. This paper is concerned with a model-theoretic study of a finite version of the canonical Ramsey theorem with a largeness condition and also a version of the Kanamori–McAloon principle. As a consequence, we produce new indicators for cuts satisfying $PA$.