Sets of Ramsey-limit points and IP-limit points

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.