On universal graphs without instances of CH. (English) Zbl 0551.03032
The author proves the striking result that a universal graph on \(\omega_ 1\) may exist under \(2^{\aleph_ 0}=\aleph_ 2\). It is assumed that in the ground model there are \(\aleph_ 2\) stationary subsets of \(\omega_ 1\) with pairwise finite intersetions, this can be achieved by Baumgartner’s thinning-out method [J. E. Baumgartner, Ann. Math. Logic 9, 401-439 (1976; Zbl 0339.04003)]. First force a graph on \(\omega_ 1\) with finite conditions, then in an iteration up to \(\omega_ 2\) the \(\alpha\) th graph will be pushed into the \(\alpha\) th stationary set. The conditions and the iteration are of finite support with some extra stipulations, similar to Shelah’s proof of Baumgartner’s theorem on \(\aleph_ 1\)-dense sets [J. E. Baumgartner, Ordered sets, Proc. NATO Adv. Study Inst., Banff/Can. 1981, 239-277 (1982; Zbl 0506.04003)]. The main part of the proof is checking ccc of \(P_{\alpha}\) by induction on \(\alpha\), along with proving that some well-behaved conditions are dense. Some generalizations are also proved.
Reviewer: P.Komjáth
References:
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| [5] | Baumgartner, J., Almost disjoint sets, the dense set problem and the partition calculus, Annals Math. Logic, 9, 401-439 (1976) · Zbl 0339.04003 |
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