Title: $\Sigma$-definability in hereditary finite superstructures and pairs of models

Corrections: A general definition of a set of relative indiscernibles should be as follows: if $M$ and $N$ are some algebraic systems then $I\subseteq M^k\cap N$ is called the set of $M$-{\it indiscernibles in} $N$ provided that $$ (M,i_0,\ldots,i_n)\equiv(M,i'_0,\ldots,i'_n) \Rightarrow (N,i_0,\ldots,i_n)\equiv(N, i'_0,\ldots,i'_n),$$ for any $i_0,\ldots,i_n,i'_0,\ldots,i'_n\in I$. In this definition, $k$ is called the {\it dimension of $I$}. Theorem 1 state that, for $c$-simple theories $T_1$ and $T_2$, if $T_2$ has an uncountable model $\Sigma$-definable over a class $Mod(T_1)$ then there exist decidable models $M$ and $N$ of $T_1$ and $T_2$, respectively, such that $N$ contains an infinite computable set of $M^*$-indiscernibles (of some finite dimension $k$), where $M^*$ is an expansion of $M$ by finitely many constants.
Definition of a wide structure should have an additional requirement: for any definable $X\subseteq M^k$ and any definable equivalence relation which divides $X$ into infinitely many classes, there exist $\bar{m}_1,\bar{m}_2\in M^{<\infty}$ s.t. this equivalence relation divides the set $\bar{m}_1\times M\times\bar{m}_2 \cap X$ into infinitely many classes also. $DLO$ and $E$ are examples of theories with wide models. In Theorems 1 and 2, if $T_1$ is a c-simple theory with wide models, then we can assume that the set of indiscernibles has dimension $1$.

PDF (Russian)

PDF (English)

PDF (corrected, Russian)