# Smooth graphs

Commentationes Mathematicae Universitatis Carolinae (1999)

- Volume: 40, Issue: 1, page 187-199
- ISSN: 0010-2628

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topSoukup, Lajos. "Smooth graphs." Commentationes Mathematicae Universitatis Carolinae 40.1 (1999): 187-199. <http://eudml.org/doc/248410>.

@article{Soukup1999,

abstract = {A graph $G$ on $\omega _1$ is called $<\!\{\omega \}$-smooth if for each uncountable $W\subset \omega _1$, $G$ is isomorphic to $G[W\setminus W^\{\prime \}]$ for some finite $W^\{\prime \}\subset W$. We show that in various models of ZFC if a graph $G$ is $<\!\{\omega \}$-smooth, then $G$ is necessarily trivial, i.eėither complete or empty. On the other hand, we prove that the existence of a non-trivial, $<\!\{\omega \}$-smooth graph is also consistent with ZFC.},

author = {Soukup, Lajos},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {graph; isomorphic subgraphs; independent result; Cohen; forcing; iterated forcing; isomorphic subgraphs; forcing; independence result; smooth graphs},

language = {eng},

number = {1},

pages = {187-199},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Smooth graphs},

url = {http://eudml.org/doc/248410},

volume = {40},

year = {1999},

}

TY - JOUR

AU - Soukup, Lajos

TI - Smooth graphs

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1999

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 40

IS - 1

SP - 187

EP - 199

AB - A graph $G$ on $\omega _1$ is called $<\!{\omega }$-smooth if for each uncountable $W\subset \omega _1$, $G$ is isomorphic to $G[W\setminus W^{\prime }]$ for some finite $W^{\prime }\subset W$. We show that in various models of ZFC if a graph $G$ is $<\!{\omega }$-smooth, then $G$ is necessarily trivial, i.eėither complete or empty. On the other hand, we prove that the existence of a non-trivial, $<\!{\omega }$-smooth graph is also consistent with ZFC.

LA - eng

KW - graph; isomorphic subgraphs; independent result; Cohen; forcing; iterated forcing; isomorphic subgraphs; forcing; independence result; smooth graphs

UR - http://eudml.org/doc/248410

ER -

## References

top- Hajnal A., Nagy Zs., Soukup L., 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. MR1117016
- Jech T., Set Theory, Academic Press, New York, 1978. Zbl1007.03002MR0506523
- Kierstead H.A., Nyikos P.J., Hypergraphs with finitely many isomorphism subtypes, Trans. Amer. Math. Soc. 312 (1989), 699-718. (1989) Zbl0725.05063MR0988883
- Shelah S., Soukup L., On the number of non-isomorphic subgraphs, Israel J. Math 86 (1994), 1-3 349-371. (1994) Zbl0797.03051MR1276143

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