Counting dependent and independent strings

The paper gives estimations for the sizes of the the following sets: (1) the set of strings that have a given dependency with a fixed string, (2) the set of strings that are pairwise \alpha independent, (3) the set of strings that are mutually \alpha independent. The relevant definitions are as follows: C(x) is the Kolmogorov complexity of the string x. A string y has \alpha -dependency with a string x if C(y) - C(y|x) \geq \alpha. A set of strings {x1, \ldots, xt} is pairwise \alpha-independent if for all i different from j, C(xi) - C(xi | xj) \leq \alpha. A tuple of strings (x1, \ldots, xt) is mutually \alpha-independent if C(x{\pi(1)} \ldots x{\pi(t)}) \geq C(x1) + \ldots + C(x_t) - \alpha, for every permutation \pi of [t].