Paper detail

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 $κ$ there is a ${\sf T}_0$ topological group of cardinality $κ$ 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 $κ$ there is a ${\sf T}_0$ topological group of cardinality $κ$ such that ONE has a winning strategy in the point-open game on the group and the group is σ-compact.