Homogeneous families and their automorphism groups
  A homogeneous family of subsets over a given set is one with a very ``rich'' automorphism group. We prove the existence of a bi-universal element in the class of homogeneous families over a given infinite set and give an explicit construction of $2^\cont$ isomorphism types of homogeneous families over a countable set. 
Menachem Kojman and Saharon Shelah