Finiteness notions induced by $\forall\exists$-sentences Throughout, work in $\mathsf{ZF}$ and restrict attention to finite languages.
Let $\mathsf{Ded}$ and $\mathsf{Amo}$ be the classes of Dedekind-finite and amorphous sets respectively. For $T$ a consistent theory with no finite models, let $\mathbb{F}_T$ be the class of sets into which no model of $T$ can be injected. I'm interested in what $\mathbb{F}_T$ can be be if $T$ is axiomatizable by a single $\forall^*\exists^*$-sentence.
Specifically, start with the following observations:

*

*By Lowenheim-Skolem we always have $\mathbb{F}_T\subseteq\mathsf{Ded}$ for appropriate $T$.


*Suppose $\varphi$ is an $\exists^*\forall^*$-sentence which is satisfiable but has no finite models. Then $\mathbb{F}_{\{\varphi\}}=\mathsf{Ded}$: if $\mathcal{M}\models\varphi$, fix some "witnessing tuple" $\overline{a}\in\mathcal{M}$ and consider the substructure of $\mathcal{M}$ generated by $\overline{a}$.


*On the other hand, letting $D$ be the theory of two disjoint infinite sets we have $\mathbb{F}_D=\mathsf{Amo}$.


*More interestingly, we have that $\mathbb{F}_T\supseteq\mathsf{Amo}$ for every satisfiable $T$ with no finite models, so the above bulletpoints represent the two extremal situations; moreover, $\mathbb{F}_{T}\supsetneq\mathsf{Amo}$ if $T$ is additionally finitely axiomatizable. (See here.)
Beyond this things are not so clear to me. In particular, the following seems natural to ask:

Is there a $\forall^*\exists^*$-sentence $\varphi$ such that $\mathbb{F}_{\{\varphi\}}$ is strictly between $\mathsf{Amo}$ and $\mathsf{Ded}$ (equivalently, such that $\mathbb{F}_{\{\varphi\}}\not=\mathsf{Ded}$)?

More generally I'm interested in understanding the quasiorder of $\forall\exists$-sentences with respect to the relation "Every model of $\varphi$ admits an injection from some model of $\psi$." This question amounts to showing that that quasiorder is nontrivial.
 A: Well. Taking a page from the "two disjoint infinite subsets" example, in the language with a function symbol and $n$ relations, we can write the sentence that the relations are pairwise disjoint and cover the whole universe, and $f$ is a function mapping each relation onto all the others, but it is not injective. This guarantees that the model is not amorphous, but we can arrange for Dedekind-finite models nonetheless.
Another more natural example would be linear order without endpoints (or at least without a maximum). Since there can be linearly ordered Dedekind-finite sets, $\Bbb F_T$ is more than just amorphous sets, but is smaller than all Dedekind-finite sets. Here, however, we have a nice independence phenomenon: in Feferman's model without free ultrafilters on $\omega$ there are Dedekind-finite sets, but every linearly orderable Dedekind-finite set is finite. So in Feferman's model $\Bbb F_T$ does in fact equal $\sf Ded$; whereas in Cohen's model it is the finite sets; and in Monro's generic extension of the Cohen model, where there is an amorphous set, it is somewhere in between.
