Is the Chang transfer $(\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\omega_2,\omega_1)$ consistent?

Foreman's 2015 Bulletin of Symbolic Logic survey of Chang's Conjecture, generic elementary embeddings, and inner models for huge cardinals highlights, among its closing open issues, the singular-cardinal Chang transfer question: Levinski, Magidor, and Shelah showed the consistency of $(\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\omega_1,\omega)$ from a huge cardinal, but it is still not known whether the stronger transfer $(\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\omega_2,\omega_1)$ is consistent relative to any large cardinal hypothesis. Later work sharpened the landscape: Eskew and Hayut proved in ZFC that the transfers with targets $(\aleph_{n+1},\aleph_n)$ for $n\geq 3$ fail, and stated that the cases with targets $(\omega_2,\omega_1)$ and $(\omega_3,\omega_2)$ are completely open; adjacent work through 2025 leaves both instances untouched. The problem is the question of whether the transfer for the $(\omega_2,\omega_1)$ target at the successor of $\aleph_\omega$ is consistent.

Background

Chang's Conjecture is an assertion about elementary substructures, written in the notation $(\kappa,\lambda)\twoheadrightarrow(\mu,\nu)$: every structure $M$ for a countable first-order language whose universe has size $\kappa$ and which carries a distinguished unary predicate $P$ interpreted as a subset of size $\lambda$ has an elementary substructure $N\prec M$ with $|N|=\mu$ and $|N\cap P|=\nu$. The classical Chang's Conjecture $(\omega_2,\omega_1)\twoheadrightarrow(\omega_1,\omega)$, formulated by C. C. Chang in two-cardinal model theory, has exact consistency strength an $\omega_1$-Erdös cardinal (Silver's theorem, as recorded with references in Foreman's survey, Theorem 1.5, and in the Kanamori–Magidor survey): an uncountable cardinal $\kappa$ is $\omega_1$-Erdös if every coloring of the finite subsets of $\kappa$ with two colors has a homogeneous subset of order type $\omega_1$. Such transfer properties are elementary consequences of the existence of certain elementary embeddings, and their consistency strength grows steeply with the cardinals involved: Kunen's construction, starting from a huge cardinal, produces transfers $(\mu^{+2},\mu^+)\twoheadrightarrow(\mu^+,\mu)$ for regular $\mu$, where a huge cardinal is a large cardinal axiom asserting the existence of an elementary embedding $j:V\to M$ with critical point $\kappa$ such that $M$ is closed under $\lambda$-sequences where $\lambda=j(\kappa)$. At successors of singular cardinals the situation changes character. Levinski, Magidor, and Shelah proved, from the consistency of a huge cardinal, that $(\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\omega_1,\omega)$ is consistent; this remains the strongest known positive transfer at the pair $(\aleph_{\omega+1},\aleph_\omega)$. The next target, $(\omega_2,\omega_1)$, has resisted all known templates: Silver-style and Shelah-style constructions at regular cardinals reach at most $(\omega_1,\omega)$-type targets, and Kunen's huge-cardinal construction requires a regular base cardinal. The obstruction has a pcf-theoretic face: pcf theory (Shelah's theory of possible cofinalities) analyzes scales, which are increasing cofinal sequences of functions in products of regular cardinals ordered modulo eventual dominance, and the existence of good points in scales places ZFC constraints on which Chang transfers can hold at successors of singulars: ZFC proves that every scale on $\aleph_\omega$ has a club of good points of cofinality at least $\omega_4$, from which $(\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\aleph_{n+1},\aleph_n)$ fails in ZFC for every $n\geq 3$ (Eskew–Hayut). Foreman's paper sits inside his program of building inner models $L[A^*,I]$ from a stationary set $A$ and the nonstationary ideal $I$ (the ideal of nonstationary subsets of a regular cardinal; a set is stationary if it meets every closed unbounded subset), using generic elementary embeddings and his decisive ideal technique to obtain models with huge cardinals; the singular transfer question, which he attributes in its difficulty to results of Cummings, is one of several issues he lists as unresolved at the close of that program. The problem stated here is that specific transfer question, not the surrounding inner-model program.

Problem Statement

Is it consistent, relative to any large cardinal hypothesis (or any consistent extension of ZFC), that the Chang transfer property $(\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\omega_2,\omega_1)$ holds — that is, that every structure for a countable language with universe of size $\aleph_{\omega+1}$ and a distinguished unary predicate of size $\aleph_\omega$ has an elementary substructure of size $\aleph_2$ whose intersection with the predicate has size $\aleph_1$? This is Foreman's formulation of the question. The companion instance with target $(\aleph_3,\aleph_2)$, which Eskew–Hayut list alongside it as completely open and which the same good-scale arguments leave untouched, is recorded as adjacent context: a solution settling both instances by the same method would be stronger, but the question asked here is the $(\omega_2,\omega_1)$ instance above, which sits at the boundary of what the pcf-theoretic obstructions leave possible.

References

1. Matthew Foreman, "Chang's Conjecture, Generic Elementary Embeddings and Inner Models for Huge Cardinals", Bulletin of Symbolic Logic 21(3) (2015) 251–269, DOI 10.1017/bsl.2015.19, https://doi.org/10.1017/bsl.2015.19
2. Jean-Pierre Levinski, Menachem Magidor, Saharon Shelah, "Chang's conjecture for $\aleph_\omega$", Israel Journal of Mathematics 69(2) (1990) 161–172, DOI 10.1007/BF02937302, https://doi.org/10.1007/BF02937302
3. James Cummings, "Collapsing successors of singulars", Proceedings of the American Mathematical Society 125(9) (1997) 2703–2709, DOI 10.1090/S0002-9939-97-03995-6, https://doi.org/10.1090/S0002-9939-97-03995-6
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5. Monroe Eskew, Yair Hayut, "On the consistency of local and global versions of Chang's Conjecture", Transactions of the American Mathematical Society 370(4) (2018) 2879–2905, DOI 10.1090/tran/7260, https://doi.org/10.1090/tran/7260 (online 2017-11-17; arXiv:1607.04904)
6. Monroe Eskew, Yair Hayut, "Global Chang's Conjecture and singular cardinals", European Journal of Mathematics 7(2) (2021) 435–463, DOI 10.1007/s40879-021-00459-8, https://doi.org/10.1007/s40879-021-00459-8 (arXiv:1812.11768)
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11. Fanxin Wu, "Two-cardinal Kurepa Hypotheses" (2025), arXiv:2510.08860, https://arxiv.org/abs/2510.08860