L. Soukup
Smooth graphs

Comment.Math.Univ.Carolinae 40,1 (1999) 187-199.

Abstract:A graph $G$ on $\omega _1$ is called $<\!{\omega }$-{smooth} if for each uncountable $W\subset \omega _1$, $G$ is isomorphic to $G[W\setminus W']$ 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.

Keywords: graph, isomorphic subgraphs, independent result, Cohen, forcing, iterated forcing
AMS Subject Classification: 03E35