The notion of minimality in model theory is related to the existence of a *gap* in the size of definable subsets of a model. Now consider the following generalization:

**Definition 1:** Let $M$ be a $\mathcal{L}$ - structure. Define:

$Def(M):=\{X\subseteq Dom(M)~|~\exists n \in \omega~~~\exists \varphi (x ,y_1 , ...,y_n)\in \mathcal{L}-Form~~~$

$\exists b_1 , ...,b_n\in Dom(M);~~~~~X=\{a\in Dom(M)~|~M\vDash \varphi(a, b_1,...,b_n)\}\}$

**Definition 2:** Let $\kappa >\lambda \geq \aleph_{0}$ be two cardinals. A model $M$ called $(\kappa , \lambda)$ - minimal if $\forall X\in Def(M);~~~~ |X|\geq \kappa~\vee~~|X|\leq \lambda$.

Assuming consistency of $\text{ZF}$:

**Question 1:** Does $\text{ZF}$ have a $(\kappa , \lambda)$ - minimal model for each $\kappa >\lambda \geq \aleph_{0}$?

If the answer of the above question is negative:

**Question 2:** Does $\text{ZF}$ have an extension like $T$ such that $T$ has a $(\kappa , \lambda)$ - minimal model for each $\kappa >\lambda \geq \aleph_{0}$?

**Question 3:** For which (probably large) cardinals like $\kappa >\lambda \geq \aleph_{0}$ does $\text{ZF}$ have a $(\kappa , \lambda)$ - minimal model?