introduced in unpublished work by Galvin, was later independently introduced by Kočinac under the name of *Rothberger boundedness*, and initially investigated in [@coc11].

In [@coc11] it was shown that a subgroup (or subset) of a metrizable topological group is Rothberger bounded if, and only if, it is strong measure zero in all left invariant metrics of the group. In [@coc11] we also extended some of the characterizations of strong measure zero from [@smzpow] to Rothberger boundedness, but we did not have techniques to also extend the Ramsey-theoretic characterization to this context. Now, in Corollary 4 of this paper, we obtain the Ramsey-theoretic characterization.

The point-open game introduced in [@Galvin] by Galvin is closely related to the notion of strong measure zero. Galvin proved that for Lindelöf spaces in which each point is an intersection of countably many open sets player ONE of the point-open game has a winning strategy if, and only if, the space is countable. Apparently few, if any, uncountable examples of Hausdorff spaces where ONE has a winning strategy in the point-open game, have been pointed out in the literature. Another objective of this paper is to show that classical work of Comfort [@Comfort] and also classical 