---
abstract: |
    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 ${\sf T}_0$ 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. (Theorem \[rothbwonbytwo\])]{}

    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. (Corollary \[rothbdedtorothbomega\])]{}

address: |
    Department of Mathematics\
    Boise State University\
    Boise, Idaho 83725
author:
- Marion Scheepers
title: Rothberger bounded groups and Ramsey theory
---

Overview
========

In topological groups boundedness properties are counterparts for covering properties in general topological spaces: Guran’s notion of $\aleph_0$-boundedness is a counterpart of the Lindelöf covering property -[@Guran], while Okunev’s notion of o-boundedness (named Menger-boundedness by Kočinac, who introduced this notion independently) and Tkachenko’s corresponding property of *strict* o-boundedness are counterparts of $\sigma$-compactness -[@CH]. In this paper we consider a boundedness property which approximates Borel’s metric notion of *strong measure zero*. This boundedness property was 