Rothberger bounded groups and Ramsey theory

We show that: (1) Rothberger bounded subgroups of sigma-compact groups are characterized by Ramseyan partition relations (Corollary 4). (2) For each uncountable cardinal kappa there is a 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 closed subspace of any sigma-compact space (Theorem 8). (3) For each uncountable cardinal kappa there is a 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 (Corollary 17). (C) 2011 Elsevier B.V. All rights reserved.