Abstract
We first prove the consistency of: there is a universal graph of power ℵ 1<2 ℵ 0 = 2 ℵ 1=ℵ 2. The consistency of the non-existence of a universal graph of power ℵ 1 is trivial. Add ℵ 2 Cohen generic reals. We then show that we can have 2 ℵ 0=ℵ 2<2 ℵ 1, and get similar results for other cardinals.
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