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On the asymptotic dimension of products of coarse spaces

We prove that for any coarse spaces $X_1,\dots,X_n$ of asymptotic dimension $\ge 1$, the product $X=X_1\times\dots\times X_n$ has asymptotic dimension $\ge n$. Another result states that a finitary coare space $Z$ has $\mathrm{asdim}(Z)\ge n$ if $Z$ admits an almost free action of the group $\mathbb Z^n$. We deduce these results from the following combinatorial result (that generalized the the Hex Theorem of Gale): for any cover $\mathcal F$ of a discrete box $K=k_1\times \dots \times k_n$, either some set $F\in\mathcal F$ contains a chain connecting two opposite faces of $K$ or there exists a set $B\subset K$ of diameter $\le 1$ such that $|\{F\in\mathcal F:F\cap B\ne\emptyset\}|>n$.