| Title:
|
Smooth graphs (English) |
| Author:
|
Soukup, L. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
1 |
| Year:
|
1999 |
| Pages:
|
187-199 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
graph |
| Keyword:
|
isomorphic subgraphs |
| Keyword:
|
independent result |
| Keyword:
|
Cohen |
| Keyword:
|
forcing |
| Keyword:
|
iterated forcing |
| MSC:
|
03E05 |
| MSC:
|
03E35 |
| idZBL:
|
Zbl 1060.03071 |
| idMR:
|
MR1715212 |
| . |
| Date available:
|
2009-01-08T18:50:58Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119073 |
| . |
| Reference:
|
[1] 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. MR 1117016 |
| Reference:
|
[2] Jech T.: Set Theory.Academic Press, New York, 1978. Zbl 1007.03002, MR 0506523 |
| Reference:
|
[3] Kierstead H.A., Nyikos P.J.: Hypergraphs with finitely many isomorphism subtypes.Trans. Amer. Math. Soc. 312 (1989), 699-718. Zbl 0725.05063, MR 0988883 |
| Reference:
|
[4] Shelah S., Soukup L.: On the number of non-isomorphic subgraphs.Israel J. Math 86 (1994), 1-3 349-371. Zbl 0797.03051, MR 1276143 |
| . |
