In this post, we give a proof that does not interpret (without parameters) in the homogeneous poset
nor in the random graph
. The idea is to first show that every continuous action of
or
on a set is faithful or trivial. We show this by using the fact that
and
have least supports. Since
and
have an automorphism of order two and
does not, this proves that there is no nontrivial continuous action of
nor
on
.