Negative universality results for graphs

S.-D. Friedman¹ ; K. Thompson²

¹ Kurt Gödel Research Center for Mathematical Logic Währinger Straße 25 A-1090 Wien, Austria
² Institut für Diskrete Mathematik und Geometrie Technische Universität Wien Wiedner Hauptstraße 8–10/104 A-1040 Wien, Austria

Fundamenta Mathematicae, Tome 210 (2010) no. 3, pp. 269-283

Résumé

It is shown that in many forcing models there is no universal graph at the successors of regular cardinals. The proof, which is similar to the well-known proof for Cohen forcing, is extended to show that it is consistent to have no universal graph at the successor of a singular cardinal, and in particular at $\aleph _{\omega +1}$. Previously, little was known about universality at the successors of singulars. Analogous results show it is consistent not just that there is no single graph which embeds the rest, but that it takes the maximal number ($2^\lambda $ for graphs of size $\lambda $) to embed the rest.

Détail

DOI : 10.4064/fm210-3-3

Keywords: shown many forcing models there universal graph successors regular cardinals proof which similar well known proof cohen forcing extended consistent have universal graph successor singular cardinal particular aleph omega previously little known about universality successors singulars analogous results consistent just there single graph which embeds rest takes maximal number lambda graphs size lambda embed rest