Sets of Ramsey-limit points and IP-limit points
Rafa{\l} Filip\'ow, Adam Kwela, Paolo Leonetti

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 be an uncountable Polish space and let be the Hindman ideal, that is, the family of all which are not -sets. For each sequence taking values in , let be the set of -limit points of . Also, let be the set of -limit points of , that is, the set of ordinary limits of subsequences with . After proving that these two notions do not coincide in general, we show that both families of nonempty sets of the type and of the type are precisely the class of nonempty analytic subsets of . 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…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Limits and Structures in Graph Theory
