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 .
Consider the following decision problem, over the equality symmetry.
Input: an orbit-finite system of linear equations over
Decide: does have a solution (not necessarily finitely-supported)?
Inspired by Eryk’s beautiful solution of this problem, we present another solution, using an amazing theorem from topological dynamics.