The consistency of 2^{aleph_{0}}> aleph_{omega} +   I(aleph_{2})=I(aleph_{omega})
  An omega-coloring is a pair <f,B> where f:[B]^{2} ---> omega. The set B is the field of f and denoted Fld(f). Let f,g be omega-colorings. We say that f realizes the coloring g if there is a one-one function k:Fld(g) ---> Fld(f) such that for all {x,y}, {u,v} in dom(g) we have f({k(x),k(y)}) not= f({k(u),k(v)}) => g({x,y}) not= g({u,v}). We write f~g if f realizes g and g realizes f. We call the ~-classes of omega-colorings with finite fields identities. We say that an identity I is of size r if |Fld(f)|=r for some/all f in I. For a cardinal kappa and f:[kappa]^2 ---> omega we define I(f) to be the collection of identities realized by f and I (kappa) to be bigcap {I(f)| f:[kappa]^2 ---> omega}.   We show that, if ZFC is consistent then ZFC + 2^{aleph_0}> aleph_omega + I(aleph_2)=I(aleph_omega) is consistent. 
Martin Gilchrist, Saharon Shelah