A non structure theorem for an infinitary theory which has the unsuperstability property

Abstract

Let $\kappa$, $\lambda$ be infinite cardinals, $\psi \in L_{\kappa^{+},\omega}$. We say that the sentence $\psi$ has the $\lambda$-unsuperstability property if there are $\{\varphi_{n}(\bar{\mathbf{x}},\bar{\mathbf{y}}): n < \omega\}$ quantifier free first order formulas in $L$, a model $M$ of $\psi$, and there exist $\{\bar{\mathbf{a}}_{\eta}: \eta \in^{\omega \geq} \lambda\} \subseteq |M|$ such that for all $\eta \in^{\omega}\lambda$, and for every $\nu \in^{\omega >}\lambda$, $$\nu < \eta \Leftrightarrow M \vDash \varphi_{l(\nu)}[\bar{\mathbf{a}}_{\nu},\bar{\mathbf{a}}_{\eta}].$$