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Keywords:
graph; isomorphic subgraphs; independent result; Cohen; forcing; iterated forcing
graph; isomorphic subgraphs; independent result; Cohen; forcing; iterated forcing
Summary:
A graph $G$ on $\omega _1$ is called $<\!{\omega}$-{\it smooth\/} if for each uncountable $W\subset \omega _1$, $G$ is isomorphic to $G[W\setminus W']$ for some finite $W'\subset W$. We show that in various models of ZFC if a graph $G$ is $<\!{\omega}$-smooth, then $G$ is necessarily trivial, i.e\. either complete or empty. On the other hand, we prove that the existence of a non-trivial, $<\!{\omega}$-smooth graph is also consistent with ZFC.
References:
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[2] Jech T.: Set Theory. Academic Press, New York, 1978. MR 0506523 | Zbl 1007.03002
[3] Kierstead H.A., Nyikos P.J.: Hypergraphs with finitely many isomorphism subtypes. Trans. Amer. Math. Soc. 312 (1989), 699-718. MR 0988883 | Zbl 0725.05063
[4] Shelah S., Soukup L.: On the number of non-isomorphic subgraphs. Israel J. Math 86 (1994), 1-3 349-371. MR 1276143 | Zbl 0797.03051
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