The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality \aleph0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are μ++ graphs on μ+ that taken jointly are universal for the graphs on μ+, while $2μ+ \gg μ++$. The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.
"Universal graphs at the successor of a singular cardinal." J. Symbolic Logic 68 (2) 366 - 388, June 2003. https://doi.org/10.2178/jsl/1052669056