Summary: Our thesis is that for the family of classes of the form EC\((T)\), \(T\) a complete first-order theory with the dependence property (which is just the negation of the independence property), there is a substantial theory which means: a substantial body of basic results for all such classes and some complementary results for the first-order theories with the independence property, as for the family of stable (and the family of simple) first-order theories. We examine some properties.
Annotated content: §1 Indiscernible sequences and averages: We consider indiscernible sequences \(\overline {\mathbf b}=\langle\overline b_t:t\in I\rangle\) wondering whether they have an average type as in the stable case. We investigate for any such \(\overline {\mathbf b}\) the set stfor\((\overline{\mathbf b})\) of formulas \(\varphi(\overline x, \overline y)\) such that every instance \(\varphi(\overline x,\overline c)\) divides \(\overline{\mathbf b}\) to a finite/co-finite set. We also consider the set dpfor\( (\overline{\mathbf b})\) of formulas \(\varphi(\overline x,\overline y)\) which can divide \(\overline{\mathbf b}\) only to finitely many intervals; this is always the case if \(T\) has the dependence property, i.e., dpfor\(( \overline{\mathbf b})=\mathbb{L}_{\tau (T)}\). If \(T\) has the dependence property, indiscernible sequences behave reasonably while indiscernible sets behave nicely. Similar behavior occurs for \(p\in{\mathcal S}(M)\) connected with the indiscernible set \(\overline{\mathbf b}\) which we call stable types. We then note the connection between unstable types, unstable \(\varphi(x,y; \overline c)\), and formulas \(\varphi(x,y;\overline c)\) with the independence property, i.e. on singletons.
§2 Characteristics of types: Each indiscernible sequence \(\overline{\mathbf b}=\langle\overline b_t:t\in I \rangle\) has for each \(\varphi=\varphi(\overline x,\overline y)\) a characteristic number \(n=n_{\overline {\mathbf b},\varphi}\), the maximal number of intervals to which an instance \(\varphi(\overline x,\overline c)\) can divide \(\overline{\mathbf b}\). We wonder what we can say about it.
§3 Shrinking indiscernibles: For an indiscernible sequence \(\langle \overline b_t:t\in I \rangle\) over a set \(A\), if we increase the set a little, i.e. if \(A'=A\cup B\), then not much indiscernability is lost. An easy case is: if \(I\) has cofinality \(>|B|+|T|\) then for some end segment \(J\) of \(I\) the sequence \(\langle\overline b_t:t\in J\rangle\) is an indiscernible sequence over \(A'\).
§4 Perpendicular endless indiscernible sequences: We define perpendicularity and investigate its basic properties; any two mutually indiscernible sequences are perpendicular. E.g., (for theories with the non-independence property) one indiscernible sequence can be perpendicular to at most \(\geq|T|^+\) pairwise perpendicular indiscernible sequences. We then deal with \({\mathbf F}^{\text{sp}}_{|T|^+}\)-constructions.
§5 Indiscernible sequences perpendicular to cuts: Using constructions as above we show that we can build models controlling quite tightly the dual cofinality of such sequences where dual-cf\((\overline {\mathbf b},M)=\text{Min}\{|B|:B\subseteq M\) and the average of \(\overline{\mathbf b}\) over \(B\cup\overline{\mathbf b}\) is not realized in \(M\}\). That is, for any pairwise perpendicular \(\langle\overline {\mathbf b}^\zeta:\zeta<\zeta^*\rangle\) we can find a model \(M\) including them with dual-cf\((\overline{\mathbf b}^\zeta,M)\) being any (somewhat large) pregiven regular cardinal).
§6 Concluding remarks: We speculate on a parallel to DOP and to deepness. Also on the existence of indiscernibles (starting with any set).