A combinatorial problem in infinite groups

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 $\mathcal{V}(w^{\ast })$ to be the class of all groups $G$ in which for any infinite subsets $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ there exist $x_i\in X_i$, $1\le i\le n$, such that $w(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)=1$. 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.