Consistency of the existence of a 4-place function on aleph_2 with no   independent subset of size k_n
  It is consistent that there is a set mapping from the four-tuples of omega_n into the finite subsets with no free subsets of size t_n for some natural number t_n. For any n< omega it is consistent that there is a set mapping from the pairs of omega_n into the finite subsets with no infinite free sets. 
Peter Komjath, Saharon Shelah