# The automorphism group of the countable atomless Boolean algebra does not have ample generics

I was told that the automorphism group of the countable atomless Boolean algebra does not have ample generics. I assume that one would show this by using the Fraisse-theoretic characterizations of countable ultrahomogeneous structures with automorphism groups with ample generics by means of finite structures expanded with $$n$$ partial automorphisms for $$n < \omega$$. However, I have not been able to find any reference for this fact. What would be a good reference? I would also be grateful for the summary of the argument if it can be sketched here.

• This is not a constructive comment, but I just find this an entertaining example of something you could put in a novel and (most) readers would not only assume that it was the mathematical equivalent of technobabble but wouldn't believe you if you said it wasn't. – Derek Elkins Apr 20 at 4:14
• This is one of the open questions listed at the end of this paper by Kechris and Rosendal. Of course, this is a well-known paper which is almost 15 years old, so I wouldn't be surprised if the question is no longer open. But that's some evidence that the proof may not be so easy. – Alex Kruckman Apr 20 at 18:19
• You might be interested in the PhD thesis of Daoud Siniora. Question 10 at the end of that thesis asks whether there is any homogeneous $\omega$-categorical structure with the strict order property which has ample generics. – Alex Kruckman Apr 20 at 18:26