Unbounded relative dimension with bounded Boolean dimension

Construct a family of posets P_n whose Boolean dimensions are uniformly bounded by a constant c while their relative dimensions tend to infinity.

Background

The paper compares relative dimension with local and Boolean dimension. It notes that there are families with bounded local dimension, and hence bounded relative dimension, but unbounded Boolean dimension; conversely, there are families with bounded Boolean dimension but unbounded local dimension. The remaining unresolved direction is whether Boolean dimension can stay bounded while relative dimension becomes unbounded.

References

Question 5: Is there a family of posets $P_n$ with $bdim(P_n) < c$ and $rdim(P_n) \to \infty$?

— Relative Dimension of Posets  (2609.05166 - Dürrschnabel et al., 4 Sep 2026) in Question 5, Section Open Problems