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.
@article{119073,
author = {Lajos Soukup},
title = {Smooth graphs},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {40},
year = {1999},
pages = {187-199},
zbl = {1060.03071},
mrnumber = {1715212},
language = {en},
url = {http://dml.mathdoc.fr/item/119073}
}
Soukup, Lajos. Smooth graphs. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) pp. 187-199. http://gdmltest.u-ga.fr/item/119073/
On the number of non-isomorphic subgraphs of certain graphs without large cliques and independent subsets, ``A Tribute to Paul Erdös '', ed. A. Baker, B. Bollobás, A. Hajnal, Cambridge University Press, 1990, pp.223-248. | MR 1117016
Set Theory, Academic Press, New York, 1978. | MR 0506523 | Zbl 1007.03002
Hypergraphs with finitely many isomorphism subtypes, Trans. Amer. Math. Soc. 312 (1989), 699-718. (1989) | MR 0988883 | Zbl 0725.05063
On the number of non-isomorphic subgraphs, Israel J. Math 86 (1994), 1-3 349-371. (1994) | MR 1276143 | Zbl 0797.03051