Sets of Ramsey-limit points and IP-limit points
Rafał Filipów, Adam Kwela, Paolo Leonetti
TL;DR
The paper investigates two notions of limit points for sequences in topological spaces: IP-limit points given by $\Lambda_x(FS)$ and Hindman-ideal limit points $\Lambda_x(H)$, showing these need not coincide in general. It then proves a striking dichotomy: for uncountable Polish spaces $X$, the families of nonempty limit-point-sets arising from these notions exhaust precisely the analytic subsets of $X$, and similarly for Ramsey convergence via $\Lambda_y(r)$ and $\Lambda_y(R)$. Moreover, the FS and Hindman frameworks piggyback on the same analytic-class realization, i.e., every nonempty analytic $C\subseteq X$ can be realized as $C=\Lambda_x(FS)=\Lambda_x(H)$ or as $C=\Lambda_y(r)=\Lambda_y(R)$, with analogous results for the Ramsey and Hindman ideals. The proofs leverage partition-regular functions, unifying IP/Ramsey-type convergence via a common scheme and extending prior work on ideal-limit-sets to these two paradigms.
Abstract
Let $X$ be an uncountable Polish space and let $\mathcal{H}$ be the Hindman ideal, that is, the family of all $S\subseteq ω$ which are not $IP$-sets. For each sequence $x=(x_n)_{n \in ω}$ taking values in $X$, let $Λ_{x}(FS)$ be the set of $IP$-limit points of $x$. Also, let $Λ_{x}(\mathcal{H})$ 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 $Λ_{x}(FS)$ and of the type $Λ_{x}(\mathcal{H})$ 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.