Smooth graphs

Lajos Soukup

Commentationes Mathematicae Universitatis Carolinae (1999)

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

Abstract

top
A graph G on ω 1 is called < ω -smooth if for each uncountable W ⊂ ω 1 , G is isomorphic to G [ W ∖ W ' ] for some finite W ' ⊂ W . We show that in various models of ZFC if a graph G is < ω -smooth, then G is necessarily trivial, i.eėither complete or empty. On the other hand, we prove that the existence of a non-trivial, < ω -smooth graph is also consistent with ZFC.

How to cite

top

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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.