3 Added definition of admits $(\kappa,\lambda)$.

From Chang and Keisler's "Model Theory", section 7.2, we know that:

1) There is a sentence $\sigma$ in a suitable language $L$ such that for all infinite cardinals $\alpha$, $\sigma$ admits $(\alpha^+,\alpha)$ iff there exists a tree $T$ of height $\alpha^+$, with at most $\alpha$ elements at each level $\xi<\alpha^+$, and with no branch of length $\alpha^+$.

2) There is a sentence $\sigma$ in a suitable language $L$ such that for all infinite cardinals $\alpha$, $\sigma$ admits $(\alpha^{++},\alpha)$ iff there is a (Kurepa) family $F$ of subsets of $\alpha^+$ such that $|F|=\alpha^{++}$ and for every $\xi<\alpha^+$, $|\{X\cap\xi|X\in F\}|=\alpha.$

My question: Are there any "natural" statements that would be equivalent to a sentence $\sigma$ admitting $(\alpha^{+n},\alpha)$, $3\le n<\omega$? For Chang and Keisler, $\sigma$ has to be first-order. For our purposes, even $L_{\omega_1,\omega}$ is good enough.

Addition: Definition A sentence $σ$ in a language with a unary predicate $P$ admits $(κ,λ)$, if $σ$ has a model $M$ such that $|M|=κ$ and $|P^M|=λ$. Of course, λ≤κ. So, not only we need a model of a specific size, but we need the predicate P to have a specific size too.

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# Is there a statement equivalent to a sentence admitting $(\alpha^{+n},\alpha)$?

From Chang and Keisler's "Model Theory", section 7.2, we know that:

1) There is a sentence $\sigma$ in a suitable language $L$ such that for all infinite cardinals $\alpha$, $\sigma$ admits $(\alpha^+,\alpha)$ iff there exists a tree $T$ of height $\alpha^+$, with at most $\alpha$ elements at each level $\xi<\alpha^+$, and with no branch of length $\alpha^+$.

2) There is a sentence $\sigma$ in a suitable language $L$ such that for all infinite cardinals $\alpha$, $\sigma$ admits $(\alpha^{++},\alpha)$ iff there is a (Kurepa) family $F$ of subsets of $\alpha^+$ such that $|F|=\alpha^{++}$ and for every $\xi<\alpha^+$, $|\{X\cap\xi|X\in F\}|=\alpha.$

My question: Are there any "natural" statements that would be equivalent to a sentence $\sigma$ admitting $(\alpha^{+n},\alpha)$, $3\le n<\omega$? For Chang and Keisler, $\sigma$ has to be first-order. For our purposes, even $L_{\omega_1,\omega}$ is good enough.