Universal graphs at the successor of a singular cardinal

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of co�nality ℵ₀. 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 2^μ+ ≫ 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 k exists. The result on the existence of universal graphs is obtained as a speci�c application of a more general method.