Asymptotic behavior of Boolean-lattice relative dimension

Characterize the asymptotic behavior of the relative dimension of the Boolean lattices B_n.

Background

The paper proves that the relative dimension of the Boolean lattice B_n tends to infinity and establishes the upper bound rdim(B_n) ≤ 2n/3 + 1/3. Its lower bound is only of order log log n divided by log log log n, leaving a substantial gap from the linear upper bound. The authors explicitly pose determining the true asymptotic growth rate as an unresolved problem.