Rothberger bounded groups and Ramsey theory

We show that:(1)Rothberger bounded subgroups of σ-compact groups are characterized by Ramseyan partition relations (Corollary 4).(2)For each uncountable cardinal κ there is a T0 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 closed subspace of any σ-compact space (Theorem 8).(3)For each uncountable cardinal κ there is a T0 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 (Corollary 17).