Sets of Ramsey-limit points and IP-limit points

2026/01/17 by Rafał Filipów, Adam Kwela, Paolo Leonetti · 1 voice
Mathematics · #math.DS #math.GN #math.LO

paper · pdf · doi:10.1090/proc/17799

Abstract

Let X be an uncountable Polish space and let H be the Hindman ideal, that is, the family of all S⊆ ω which are not IP-sets. For each sequence x=(x_n)_{n ∈ ω} taking values in X, let Λ_x(FS) be the set of IP-limit points of x. Also, let Λ_x(H) be the set of H-limit points of x, that is, the set of ordinary limits of subsequences (x_n)_{n ∈ S} with S∉ 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(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.

Citations

• Borel complexity of sets of ideal limit points
• Topological complexity of ideal limit points
• A unified approach to Hindman, Ramsey and van der Waerden spaces
• More on ideal limit points
• The Borel complexity of ideal limit points
• Strongly mixing systems are almost strongly mixing of all orders
• On the Relationship between Ideal Cluster Points and Ideal Limit Points
• On the relationship between ideal cluster points and ideal limit points
• Invariance of Ideal Limit Points
• Invariance of ideal limit points

Discussions

• Rafa{\l} Filip\'ow, Adam Kwela, Paolo Leonetti: Sets of Ramsey-limit points and IP-limit points https://arxiv.org/abs/2601.12187 https://arxiv.org/pdf/2601.12187 https://arxiv.org/html/2601.12187 [bsky, 0 points, 0 comments]

Related

• On the Relationship between Ideal Cluster Points and Ideal Limit Points
• Critical partition regular functions for compact spaces
• Invariance of Ideal Limit Points
• Rough families, cluster points, and cores
• Limit points of subsequences