Classifying homogeneous ultrametric spaces up to coarse equivalence

For every metric space ▫$X$▫ we introduce two cardinal characteristics ▫${rm cov}^flat(X)$▫ and ▫${rm cov}^sharp(X)$▫ describing the capacity of balls in ▫$X$▫. We prove that these cardinal characteristics are invariant under coarse equivalence and prove that two ultrametric spaces ▫$X,Y$▫ are coarsely equivalent if ▫${rm cov}^flat(X)={rm cov}^sharp(X)={rm cov}^flat(Y)={rm cov}^sharp(Y)$▫. This result implies that an ultrametric space ▫$X$▫ is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if ▫${rm cov}^flat(X)={rm cov}^sharp(X)$▫. Moreover, two isometrically homogeneous ultrametric spaces ▫$X,Y$▫ are coarsely equivalent if and only if ▫${rm cov}^sharp(X)={rm cov}^sharp(Y)$▫ if and only if each of these spaces coarsely embeds into the other space. This means that the coarse structure of an isometrically homogeneous ultrametric space ▫$X$▫ is completely determined by the value of the cardinal ▫${rm cov}^sharp(X)={rm cov}^flat(X)$▫