Sets of Ramsey-limit points and IP-limit points

TL;DR

This paper investigates the relationship between Ramsey-limit points and IP-limit points in uncountable Polish spaces, showing both sets of limit points form exactly the nonempty analytic subsets, using partition regular functions.

Contribution

It establishes that both types of limit points are precisely the nonempty analytic subsets of the space, and introduces a unified approach via partition regular functions.

Findings

Both families of limit points are exactly the nonempty analytic subsets.

The notions of IP-limit points and Ramsey-limit points do not coincide in general.

Partition regular functions provide a unified framework for these convergence types.

Abstract

Let $X$ be an uncountable Polish space and let $\mathcal{H}$ be the Hindman ideal, that is, the family of all $S\subseteq \omega$ which are not $IP$-sets. For each sequence $x=(x_n)_{n \in \omega}$ taking values in $X$, let $\Lambda_{x}(FS)$ be the set of $IP$-limit points of $x$. Also, let $\Lambda_{x}(\mathcal{H})$ be the set of $\mathcal{H}$-limit points of $x$, that is, the set of ordinary limits of subsequences $(x_n)_{n \in S}$ with $S\notin \mathcal{H}$. After proving that these two notions do not coincide in general, we show that both families of nonempty sets of the type $\Lambda_{x}(FS)$ and of the type $\Lambda_{x}(\mathcal{H})$ are precisely the class of nonempty analytic subsets of $X$. An analogous result holds also for Ramsey convergence. In the proofs, we use the concept of partition regular functions introduced in J. Symb. Log. (2024) [doi:10.1017/jsl.2024.8], which provide a unified approach to these types of convergence.