Letting $\alpha$ be an infinite cardinal and $3\leq n\lt\omega$,
I think Peter Komjath's proposed statement $2^\alpha\geq\alpha^{+n}$ is the simplest and most natural equivalent of a first-order sentence admitting $(\alpha^{+n},\alpha)$: just let $\sigma$ say a binary relation is extensional with domain given by a predicate and range equal to the universe. However, you asked for a GCH example in your comment, so I suggest the following statement, that a generalized kind of Kurepa family exists.

- $(*_{\alpha,n})$ says there exists $\mathcal{X}\subset\mathcal{P}(\alpha^{+n-1})$ of size $\alpha^{+n}$ such that $|\{X\cap A: X\in\mathcal{X}\}|\leq\alpha$ for all $A\subset\alpha^{+n-1}$ of size $\leq\alpha$.

To get our first-order sentence $\sigma$, we use the fact that $[\alpha^{+n-1}]^{\leq\alpha}$ has nice cofinal subsets, and the equivalence of $(*_{\alpha,n})$ to its formal weakenings where $A$ is quantified over a cofinal set. Our first-order sentence $\sigma$ says that

- the universe is $L_n$,
- $(L_0,\lt_0),\ldots,(L_n,\lt_n)$ are linear orders,
- $f_i(x,\bullet)\colon L_i \rightarrow \{ y : y\leq_{i+1} x\}$ is always onto,
- $g(x,\bullet)\colon L_{n-1}\rightarrow \{0,1\}$ for all $x\in L_n$,
- $g(x,\bullet)\not=g(y,\bullet)$ for all $x\not= y$,
- $h(x_{n-1},\ldots,x_0,\bullet)\colon L_0\rightarrow 2$
for all $x\in\prod_{i=0}^{n-1} L_i$, and
- every $g(x_n,f_{n-2}(x_{n-1},f_{n-3}(x_{n-2},\cdots,f_0(x_1,\bullet)\cdots)))$
equals some $h(x_{n-1},\ldots,x_0,\bullet)$.

$(*_{\alpha,n})$ holds iff $\sigma$ has a model with size $\alpha^{+n}$ with $L_0$ of size $\alpha$.

Jensen proved that something stronger than $(*_{\alpha,n})$ holds if $V=L$.

- KH($\kappa,\lambda$) says that there exists $\mathcal{F}\subset\mathcal{P}(\kappa)$ of size $\kappa^+$ such that $|\{X\cap A:X\in\mathcal{F}\}|\leq|A|+\aleph_0$ for all $A\subset\kappa$ of size $\lt\lambda$.

Clearly, KH($\alpha^{+n-1},\alpha^+$) implies $(*_{\alpha,n})$.

Jensen proved that if $V=L$, then KH($\kappa,\lambda$) holds for all regular uncountable $\kappa$ and all uncountable $\lambda<\kappa$. In particular, $(*_{\alpha,n})$ is always true in $L$. Jensen proved that $V=L$ also implies KH($\kappa,\kappa$) for all regular uncountable cardinals that are not ineffable.
Jensen and Kunen proved that if $\kappa$ is ineffable,
then KH($\kappa,\kappa$) fails. Devlin's exposition of the proofs
is available here; see the chapter "Ineffable cardinals and the generalised Kurepa hypothesis."

On the other hand, $(*_{\alpha,n})$ is directly refuted by the Chang conjecture variant $(\alpha^{+n},\alpha^{+n-1})\twoheadrightarrow(\alpha^+,\alpha)$.
For regular $\alpha$, we can force this Chang conjecture easily using the method of Levinski-Magidor-Shelah (MR1045371).
Assume GCH and epsilon more than a huge embedding: $j\colon V\prec N\supset {}^{\lambda^+}N$ where $\kappa=cp(j)$ and $\lambda=j(\kappa)$. Since $N$ knows that $j''\mathfrak{A}\prec j(\mathfrak{A})$ for all structures $\mathfrak{A}$ of the form $(H(\lambda^+),\in,P)$, we have $(\lambda^+,\lambda)\twoheadrightarrow (\kappa^+,\kappa)$ in $V$ by elementarity of $j$.
The two-step iteration $\mathbb{P}=\mathrm{Coll}(\alpha,\kappa)*\mathrm{Coll}(\kappa^{+n-2},\lt\lambda)$ preserves GCH and forces $(\alpha^{+n},\alpha^{+n-1})\twoheadrightarrow(\alpha^+,\alpha)$. The key point is that because there are only $\lambda$-many nice $\mathbb{P}$-names for elements of $\lambda$, this set of names Chang-transfers to a $\kappa$-sized set of names for elements of $\lambda$, implying our desired transfer from $\alpha^{+n-1}$ to $\alpha$ in the generic extension $V[G]$:
if in $V$ we have

- $\mathbb{P}\in M\prec(H(\lambda^+),\in,\dot{P})$,
- $|M|=\kappa^+$, and
- $|M\cap\lambda|=\kappa$,

then in $V[G]$ we have

- $M[G]\prec(H(\lambda^+),\in,\dot{P}_G)=(H(\alpha^{+n}),\in,\dot{P}_G)$,
- $|M[G]|=\kappa^+=\alpha^+$, and
- $|M[G]\cap\alpha^{+n-1}|=|M[G]\cap\lambda|=|\kappa|=\alpha$.

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