Negative universality results for graphs
Volume 210 / 2010
Fundamenta Mathematicae 210 (2010), 269-283
MSC: Primary 03E75, 03E35; Secondary 03E05, 03E55.
DOI: 10.4064/fm210-3-3
Abstract
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.
