strategy in the point-open game if, and only if, ONE has a winning strategy in ${{\sf G}_1}({\mathcal{O}},{\mathcal{O}})$.

A Ramseyan characterization of some Rothberger bounded groups.
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Let $\mathcal{A}$ and $\mathcal{B}$ be families of sets and let $n$ and $k$ be positive integers. The symbol $$\mathcal{A}\rightarrow(\mathcal{B})^n_k$$ denotes the statement that for each $A\in\mathcal{A}$ and for each function $f:\lbrack A\rbrack^n\rightarrow\{1,\,\cdots,\, k\}$ there is a $B\subseteq A$ and an $i\in\{1,\,\cdots,\,k\}$ such that $B\in\mathcal{B}$ and $f$ is constant of value $i$ on $\lbrack B\rbrack^n$.

An open cover $\mathcal{U}$ of $Y$ is said to be an $\omega$-cover if $Y\not\in\mathcal{U}$, but for each finite set $F\subseteq Y$ there is a $U\in\mathcal{U}$ with $F\subseteq U$. $\Omega$ denotes the family of all open $\omega$-covers of $Y$.

Theorem 9 of [@smzpow] gives the following characterization of strong measure zero subsets of $\sigma$-compact metric spaces:

\[smzpowTh9\] For $X$ a subspace of a $\sigma$-compact metric space $Y$ the following are equivalent:

1.  [$Y$ has the property ${{\sf S}_1}({\mathcal{O}},\mathcal{O}_X)$.]{}

2.  [$X$ has strong measure zero (in all equivalent metrics on $Y$).]{}

3.  [ONE has no winning strategy in the game ${{\sf G}_1}({\mathcal{O}},\mathcal{O}_X)$.]{}

4.  [For each positive integer $k$, $\Omega\rightarrow({\mathcal{O}}_X)^2_k$.]{}

Failure of Lebesgue’s Covering Lemma (which holds for compact metrizable spaces) was the 