On the asymptotic dimension of products of coarse spaces
Banakh, Iryna ; Banakh, Taras, 1968-
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 ▫$\mathscr F$▫ of a discrete box ▫$K=k_1\times \dots \times k_n$▫, either some set ▫$F\in\mathscr F$▫ contains a chain connecting two opposite faces of ▫$K$▫ or there exists a set ▫$B\subset K$▫ of diameter ▫$\le 1$▫ such that ▫$|\{F\in\mathscr F:F\cap B\ne\emptyset\}|>n$▫.
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