Rothberger bounded groups and Ramsey theory

We show that: 1. Rothberger bounded subgroups of sigma-compact groups are characterized by Ramseyan partition relations. 2. For each uncountable cardinal $\kappa$ there is a ${\sf T}_0$ topological group of cardinality $\kappa$ such that ONE has a winning strategy in the point-open game on the group and the group is not a subspace of any sigma-compact space. 3. For each uncountable cardinal $\kappa$ there is a ${\sf T}_0$ topological group of cardinality $\kappa$ such that ONE has a winning strategy in the point-open game on the group and the group is \sigma-compact.

💡 Research Summary

The paper investigates the interplay between three seemingly disparate topics—Rothberger boundedness, Ramsey‐type partition relations, and infinite‐length topological games—within the setting of topological groups. A Rothberger bounded space is one that satisfies the strong selection principle S₁(𝒪,𝒪): for every sequence of open covers (𝒰ₙ)ₙ∈ℕ there exist single members Uₙ∈𝒰ₙ such that {Uₙ : n∈ℕ} still covers the whole space. This property is known to be equivalent to ONE having a losing strategy in the point‑open game G₁(𝒪,𝒪), where ONE chooses points and TWO responds with an open set containing the point; ONE wins if TWO’s chosen open sets eventually cover the space. The authors turn this equivalence around: they ask what structural features a topological group must possess for ONE to have a winning strategy, i.e., for the group to fail the Rothberger property.

The first major result provides a precise characterization of Rothberger bounded subgroups of σ‑compact groups. Let G be a σ‑compact topological group and H≤G a subgroup. The authors prove that H is Rothberger bounded if and only if for every countable ordinal α<ω₁ the classical Ramsey partition relation \