On the asymptotic dimension of products of coarse spaces

We prove that for any coarse spaces X-1, ..., X-n of asymptotic dimension >= 1, the product X = X-1 x ... xX(n) has asymptotic dimension >= n. Another result states that a finitary coarse space Zhas asdim(Z) >= n if Zadmits an almost free action of the group Z(n). We deduce these inequalities from the following combinatorial result (that generalizes the Hex Theorem of Gale): for any cover Fof a discrete box K= k(1) x ... xk(n), either some set F. Fcontains a chain connecting two opposite faces of Kor there exists a set B subset of K of diameter <= 1that intersects at least n + 1 elements of the cover F. (C) 2021 Elsevier B.V. All rights reserved.