  Let $w$ be a word in the free group of rank $n \in \mathbb{N}$ and let
$\mathcal{V}(w)$ be the variety of groups defined by the law $w=1$. Define
$\mathcal{V}(w^*)$ to be the class of all groups $G$ in which for any infinite
subsets $X_1, ..., X_n$ there exist $x_i \in X_i$, $1\leq i\leq n$, such that
$w(x_1, ..., x_n)=1$. Clearly, $\mathcal{V}(w) \cup \mathcal{F} \subseteq
\mathcal{V}(w^*)$; $\mathcal{F}$ being the class of finite groups. In this
paper, we investigate some words $w$ and some certain classes $\mathcal{P}$ of
groups for which the equality $(\mathcal{V}(w) \cup \mathcal{F})\cap
\mathcal{P}= \mathcal{P} \cap \mathcal{V}(w^*)$ holds.
