Rothberger bounded groups and Ramsey theory

TL;DR

This paper explores the relationship between Rothberger bounded groups and Ramsey theory, characterizing certain subgroups via partition relations and constructing topological groups with specific game-theoretic properties.

Contribution

It establishes a characterization of Rothberger bounded subgroups using Ramseyan partition relations and constructs topological groups with particular winning strategies in the point-open game.

Findings

Rothberger bounded subgroups are characterized by Ramseyan partition relations.

Existence of uncountable topological groups where ONE wins the point-open game but the group is not sigma-compact.

Existence of uncountable topological groups where ONE wins the point-open game and the group is sigma-compact.

Abstract

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.