The so-called structural Ramsey theorem for finite Boolean algebras was proved by Graham and Rothschild in a paper Ramsey's Theorem for $n$-parameter sets, or so I am told. A problem that I'm having reading the paper is that the said theorem is stated and proved in terms of $n$-parameter sets, with which I am not familiar or comfortable (a single $n$-parameter set is determined by seven parameters!).
Is there a proof of the structural Ramsey theorem for finite Boolean algebras written in terms of Boolean algebras?
Addendum: the structural Ramsey theorem that I have in mind is that the category of finite Boolean algebras and embeddings have the Ramsey property as defined in the aforementioned Wikipedia article.