Superlimit

Then the following are equivalent: 1) For all high-enough #92;lambda, there exists #92;kappa#92;le#92;lambda such that there is a good #92;lambda-frame on the class of #92;kappa-saturated models in K#95;#92;lambda. 2) For all high-enough #92;lambda, K has a unique limit model of cardinality #92;lambda. 3) For all high-enough #92;lambda, K has a superlimit model of cardinality #92;lambda. 4) For all high-enough #92;lambda, the union of a chain of #92;lambda-saturated models is #92;lambda-saturated. 5) There exists #92;theta such that for all high-enough #92;lambda, K is (#92;lambda,#92;theta)-solvable..