Geodesic
Smooth graphs
Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 1, pp. 187-199 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 03E05, 03E35
Keywords: graph; isomorphic subgraphs; independent result; Cohen; forcing; iterated forcing 
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     author = {Soukup, L.},
     title = {Smooth graphs},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {187--199},
     year = {1999},
     volume = {40},
     number = {1},
     mrnumber = {1715212},
     zbl = {1060.03071},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1999_40_1_a14/}
}
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Soukup, L. Smooth graphs. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 1, pp. 187-199. http://geodesic.mathdoc.fr/item/CMUC_1999_40_1_a14/