Big Ramsey degrees of the countable atomless Boolean algebra

Determine whether the countable atomless Boolean algebra has finite big Ramsey degrees and whether it admits a big Ramsey structure.

Background

Although finite antilexicographically ordered Boolean algebras satisfy a Ramsey theorem via the dual Ramsey theorem, the corresponding infinitary big Ramsey problem for the countable atomless Boolean algebra remains unresolved.

References

Does the countable atomless Boolean algebra have finite big Ramsey degrees? Does it admit a big Ramsey structure?

— Twenty years of Nešetřil's classification programme of Ramsey classes  (2501.17293 - Hubička et al., 28 Jan 2025) in Question, subsection Big Ramsey structures