### 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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