What is the dimension of a subspace of the product of $n$ linearly ordered compacta This question is motivated by this problem of Dominic van der Zypen.
Problem. Let $X=\prod_{i=1}^nX_i$ be the Tychonoff product of linearly ordered compact Hausdorff spaces $X_1,\dots,X_n$. Is it true that each subspace $Y\subset X$ has large inductive dimension $\mathrm{Ind}(Y)\le n$ and the covering dimension $\mathrm{dim}(Y)\le n$?
Remark 1. By induction it can be shown that each subspace $Y$ of $X$ has small inductive dimension $\mathrm{ind}(Y)\le n$. 
 A: In an old PhD-thesis "Finite products of locally compact ordered spaces" by J. van Dalen (Vrije Universiteit, Amsterdam) from 1972, I found (I could not find a paper with the result, so far, as I have no access to a university library, but I have the thesis on my shelves) the following Corollary 12.1 (page 45): 

Let $m$ be a natural number and let $X_1, \ldots, X_m$ be non-degenerate connected ordered spaces. Then $\operatorname{ind}(\prod_{i=1}^m X_i)= m$.

And corollary 12.2 , which has the same data plus the extra assumption that $\prod_{i=1}^m X_i$ is normal (to make $\operatorname{Ind}$ sensibly defined, I suppose) and concludes that $\dim(\prod_{i=1}^m X_i) = \operatorname{Ind}(\prod_{i=1}^m X_i) = m$ as well.
In the references I also found
I.K. Lifanov, "Dimensionality of the product of ordered continua", Dokl. Akad. Nauk SSSR 177 (1967), 778-781 (Sov. Math. Dokl. 8 (1967), 1500-1503)
and 
I.K. Lifanov, "The dimension of a product of unidimensional bicompacta", Dokl. Akad. Nauk SSSR 180 (1968), 534-537 (Sov. Math. Dokl. 9 (1968), 648-651)
which look relevant too.
A: (All spaces are assumed to be Hausdorff). For every closed subspace $A$ of an arbitrary compact space $X$, we have:
$$ \dim(A)\ \le\ \dim(X) $$
This more than answers one of the questions. (The other q. would make me think, ouch!).
