Unions of cubes in $\mathbb{R}^{n}$, combinatorics in $\mathbb{Z}^{n}$ and the John-Nirenberg and John-Str\"omberg inequalities

Suppose that the $d$-dimensional unit cube $Q$ is the union of three disjoint "simple" sets $E$, $F$ and $G$ and that the volumes of $E$ and $F$ are both greater than half the volume of $G$. Does this imply that, for some cube $W$ contained in $Q$. the volumes of $E\cap W$ and $F\cap W$ both exceed $s$ times the volume of $W$ for some absolute positive constant $s$? Here, by "simple" we mean a set which is a union of finitely many dyadic cubes. We prove that an affirmative answer to this question would have deep consequences for the important space $BMO$ of functions of bounded mean oscillation introduced by John and Nirenberg. The notion of a John-Str\"omberg pair is closely related to the above question, and the above mentioned result is obtained as a consequence of a general result about these pairs. We also present a number of additional results about these pairs. (The second and third versions present the same results as the first version. The bibliography has been updated. The presentation is more detailed and hopefully more reader-friendly. Some misprints and some small errors in a couple of the proofs have been corrected.)