Classifying homogeneous cellular ordinal balleans up to coarse equivalence

For every ballean ▫$X$▫ we introduce two cardinal characteristics ▫$text{cov}^flat(X)$▫ and ▫$text{cov}^sharp(X)$▫ describing the capacity of balls in ▫$X$▫. We observe that these cardinal characteristics are invariant under coarse equivalence and prove that two cellular ordinal balleans ▫$X,Y$▫ are coarsely equivalent if ▫$text{cof}(X)=text{cof}(Y)$▫ and ▫$text{cov}^flat(X) = text{cov}^sharp(X) = text{cov}^flat(Y) = text{cov}^sharp(Y)$▫. This result implies that a cellular ordinal ballean ▫$X$▫ is homogeneous if and only if ▫$text{cov}^flat(X)=text{cov}^sharp(X)$▫. Moreover, two homogeneous cellular ordinal balleans ▫$X,Y$▫ are coarsely equivalent if and only if ▫$text{cof}(X)=text{cof}(Y)$▫ and ▫$text{cov}^sharp(X) = text{cov}^sharp(Y)$▫ if and only if each of these balleans coarsely embeds into the other ballean. This means that the coarse structure of a homogeneous cellular ordinal ballean ▫$X$▫ is fully determined by the values of the cardinals ▫$text{cof}(X)▫$ and ▫$text{cov}^sharp(X)$▫. For every limit ordinal ▫$gamma$▫ we shall define a ballean ▫$2^{<gamma}$▫ (called the Cantor macro-cube), which in the class of cellular ordinal balleans of cofinality ▫$text{cf}(gamma)$▫ plays a role analogous to the role of the Cantor cube ▫$2^{kappa}$▫ in the class of zero-dimensional compact Hausdorff spaces. We shall also present a characterization of balleans which are coarsely equivalent to ▫$2^{