  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.
