Universal graphs between a strong limit singular and its power

The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper G bold upper C bold upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">G</mml:mi> <mml:mi mathvariant="bold">C</mml:mi> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {GCH}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi> κ </mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is supercompact and the cardinals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta greater-than kappa"> <mml:semantics> <mml:mrow> <mml:mi> θ </mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi> κ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\theta &gt; \kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda greater-than kappa"> <mml:semantics> <mml:mrow> <mml:mi> λ </mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi> κ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda &gt; \kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are regular, as an application of a more general method, we obtain a forcing extension in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c f left-parenthesis kappa right-parenthesis equals theta"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> κ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi> θ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">cf(\kappa ) = \theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the Singular Cardinal Hypothesis fails at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi> κ </mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and there exists a universal graph at cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda element-of left-parenthesis kappa comma 2 Superscript kappa Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> λ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi> κ </mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi> κ </mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda \in (\kappa ,2^\kappa )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .