Hajnal, András; Nagy, Z.; Soukup, L. On the number of certain subgraphs of graphs without large cliques and independent subsets. (English) Zbl 0738.05063 A tribute to Paul Erdős, 223-248 (1990). [For the entire collection see Zbl 0706.00007.]H. Kierstead and P. Nyikos proved that if an \(n\)-uniform hypergraph \(G\) on \(\alpha\) vertices is isomorphic to each of its induced subgraphs on \(\alpha\) vertices then \(G\) is either empty nor complete. In the paper under review it is shown that if \(G\) is a graph on \(\omega_ 1\) which is neither complete nor empty then the number of all isomorphism classes of induced subgraphs of \(G\) on \(\omega_ 1\) is at least \(2^ \omega\). In addition a construction of a non-trivial graph \(G\) on \(\omega_ 1\) with the property that \(G\) is isomorphic to \(G[W]\) whenever \(W\subset V(G)\) with \(| V(G)\backslash W|<| V(G)|\). Reviewer: P.Horák (Bratislava) Cited in 2 Documents MSC: 05C65 Hypergraphs 05C35 Extremal problems in graph theory 03E10 Ordinal and cardinal numbers Keywords:induced subgraphs Citations:Zbl 0706.00007 PDFBibTeX XMLCite \textit{A. Hajnal} et al., in: A tribute to Paul Erdős. Cambridge etc.: Cambridge University Press. 223--248 (1990; Zbl 0738.05063)