On page 241 of William Reinhardt's paper "[Ackermann's set theory equals ZF](https://www.sciencedirect.com/science/article/pii/0003484370900112)" (Annals of math. Logic vol. 2, 1970), question 4.15 is the following:
> How large is the first ordinal $\gamma$ such that there are ordinals $\alpha_i$, $i\in\omega$ for which $i<j\rightarrow\alpha_i<\alpha_j$ and
>
> 3) $(V_\gamma,\in,\alpha_i)_{i\in\omega}\equiv(V_\gamma,\in,\alpha_{i+1})_{i\in\omega}$?
Is the answer to this question now known?
Here $V_\gamma$ is the $\gamma$th level of the [von Neumann hierarchy](https://en.wikipedia.org/wiki/Von_Neumann_universe) and $\equiv$ denotes elementary equivalence. I am reading the structures as $(V_\gamma,\in,\alpha_0,\alpha_1,\alpha_2,\ldots)$ and $(V_\gamma,\in,\alpha_1,\alpha_2,\alpha_3,\ldots)$.
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Some existing work: these ordinals $\gamma$ appear to be such that $V_\gamma$ has a set of indiscernibles of order type $\omega$ (although I can't prove this - in the next section of this post there is a related property 3').) By lemma 1 of F. R. Drake's "[A fine hierarchy of partition cardinals](https://eudml.org/doc/214644)" (Fundamenta Mathematicae vol. 81, iss. 3, 1974), if $\gamma$ is such that $\gamma\rightarrow(\omega)^{<\omega}_{2^{\aleph_0}}$, then $\gamma$ has property 3). Reinhardt remarks that via partition properties, if $\gamma$ is measurable then $\gamma$ has property 3). However Reinhardt says "But the condition 3) is probably quite weak". I was unable to produce any sequence $(\alpha_i)_{i\in\omega}$ for some small $\gamma$, e.g. the least inaccessible.
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If no complete answer is known, some of my work: I do not know how to deal with $\gamma$ with property 3) that may not be cardinals, but I have been trying to find partition properties implied by 3) as it looks like an indiscernibility property. My current progress is on a similar principle 3'), although I was unable to prove it equivalent to 3):
> There are ordinals $\alpha_i$, $i\in\omega$ for which $i<j\rightarrow\alpha_i<\alpha_j$ and
>
> 3') For any finite sequences $i_0<i_1<\ldots<i_n$ and $j_0<j_1<\ldots<j_n$ of natural numbers, $(V_\gamma,\in,\alpha_{i_0},\alpha_{i_1},\ldots,\alpha_{i_n})\equiv(V_\gamma,\in,\alpha_{j_0},\alpha_{j_1},\ldots,\alpha_{j_n})$.
I believe I have proved that $\gamma$ is a cardinal and has property 3'), then $\gamma$ satisfies a definable partition principle which might be called "$\gamma\overset{\Sigma_\omega}{\rightarrow}(\omega)^{<\omega}_{2^{\aleph_0}}$", namely for all $f:[\gamma]^{<\omega}\to 2^{\aleph_0}$ such that $f$ is parameter-free definable on $(V_\gamma,\in)$, there is a $Y\subseteq\gamma$ of order type $\omega$ which is homogeneous for $f$.
This is proved by modifying the reverse direction of theorem 2.1 in ch.8, §2 of Drake's *Set Theory: An Introduction to Large Cardinals* (1974) to use only the structure $\mathfrak A=(V_\gamma,\in)$ in place of structures $\mathfrak A=(\gamma,<,(R_{n,\beta})_{n,\beta<\omega})$, as property 3') does not include extra relation symbols like the book's indiscernibility property does. As $f$ is definable on $(V_\gamma,\in)$, for each relation $R_{n,\beta}$ there is a formula $\rho_{n,\beta}$ in the unaugmented language such that $R_{n,\beta}(x_1,\ldots,x_n)$ iff $(V_\gamma,\in)\vDash\rho_{n,\beta}(x_1,\ldots,x_n)$ for any $x_1,\ldots,x_n\in V_\gamma$. Then once we extract the set $(Y,<)$ of indiscernibles, for all $n,\beta<\omega$ and length-$n$ sequences $x_1<\ldots<x_n$, $x'_1<\ldots<x'_n$ from $Y$, $(V_\gamma,\in)\vDash\rho_{n,\beta}(x_1,\ldots,x_n)$ iff $(V_\gamma,\in)\vDash\rho_{n,\beta}(x'_1,\ldots,x'_n)$ as usual, so $Y$ is homogeneous for $f$.
Definable versions of partition properties have been considered before, e.g. in Bagaria and Bosch's "[Generic absoluteness under projective forcing](https://www.impan.pl/shop/publication/transaction/download/product/88246)" (Fundamenta Mathematicae 194, pp.95--120, 2007) the property $\kappa\overset{\Sigma_\omega}{\rightarrow}(\kappa)^2$ is considered. Qi Feng's "[A hierarchy of Ramsey cardinals](https://www.sciencedirect.com/science/article/pii/016800729090028Z)" (Annals of Pure and Applied Logic vol. 49, 1990) defines $\Pi_\alpha$-Ramsey cardinals for ordinal $\alpha$, but even for $\alpha\in\omega$ I am not aware of any correspondence to a principle for $\Pi_\alpha$-definable partitions.