The Consistency of ${\rm ZFC}+2^{\aleph_{0}}>\aleph_{\omega}+ {\cal I}(\aleph_2)={\cal I}(\aleph_{\omega})$

by Gilchrist and Shelah. [GcSh:583]
J Symbolic Logic, 1997
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.

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