$\Sigma$-Definability in Hereditarily Finite Superstructures and Pairs of Models

We consider the problem of being $\Sigma$-definable for an uncountable model of a $c$-simple theory in hereditarily finite superstructures over models of another $c$-simple theory. A necessary condition is specified in terms of decidable models and the concept of relative indiscernibility introduced in the paper. A criterion is stated for the uncountable model of a $c$-simple theory to be $\Sigma$-definable in superstructures over dense linear orders, and over infinite models of the empty signature. We prove the existence of a $c$-simple theory (of an infinite signature) every uncountable model of which is not $\Sigma$-definable in superstructures over dense linear orders. Also, a criterion is given for a pair of models to be recursively saturated.