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 $x_{0},\ldots,x_{n-1}$ and $y_{0},\ldots,y_{n-1}$ in $H$, $f=f$ 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 $f:[\mathbb{N}]^{n}\rightarrow\mathbb{N}$, 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 $\operatorname{PA}$