Let $X_1,\ldots,X_p$ be random variables on the same space. Define their dependence rank, denoted $rank(X_1,\ldots,X_p)$ as the largest nonnegative integer $k$ such that there is a subcollection of $k$ out of the $p$ random variables which are pairwise statistically independent (i.e theire joint distribution funciton factors as a product of their individual distribution funcitons). Of course, every singleton $\{X_i\}$ forms an independent collection. Thus $1 \le rank(X_1,\ldots,X_p) \le p$, and the upper bound is attained $X_1,\ldots,X_p$ are independent to begin with.

Question. Is the above notion of rank defined above studied in the literature ? Are the any known estimates for it as function of, say, the covariance matrix of $\Sigma$ of $X_1,\ldots,X_p$ ?