A set of conjectures about Ann is a subset \(\C_A\subseteq \pow(T_A)\) (similarly, the set of conjectures about Bob is a subset \(\C_B\subseteq \pow(T_B)\)). A structure \(\langle \{T_A, T_B\}, \{\lambda_A, \lambda_B\}\rangle\) is said to be assumption-complete for the conjectures \(\C_A\) and \(\C_B\) provided for each conjecture in \(\C_A\) there is a type that assumes that conjecture (similarly for Bob). Formally, for each \(Y\in\C_B\) there is a \(t_0\in T_A\) such that \(\lambda_A(t_0)=Y\), and similarly for Bob. As we remarked above, a simple counting argument shows that when \(\C_A=\pow(T_A)\) and \(\C_B=\pow(T_B)\), then assumption-complete models only exist in trivial cases. A much deeper result is: