# Chang's conjecture

Chang's conjecture is a model theoretic assertion which implies many structures of a certain variety have elementary substructures of another variety. Chang's conjecture was originally formulated in 1963 by Chen Chung Chang and Vaught.

Chang's conjecture is equiconsistent over $\text{ZFC}$ to the existence of the $\omega_1$-Erdős cardinal. In particular, if you collapse an $\omega_1$-Erdős cardinal to $\omega_2$ with the Silver collapse, then in the resulting model Chang's conjecture holds. On the other hand, if Chang's conjecture is true, then $\omega_2$ is $\omega_1$-Erdős in a transitive inner model of $\text{ZFC}$. 

Chang's conjecture implies $0^{\#}$ exists. 

## Definition

The notation $(\kappa,\lambda)\twoheadrightarrow(\nu,\mu)$ is the assertion that every structure $\mathfrak{A}=(A;R^A...)$ with a countable language such that $|A|=\kappa$ and $|R^A|=\lambda$ ( a model of "type $(\kappa,\lambda)$) has a proper elementary substructure $\mathfrak{B}=(B;R^B...)$ with $|B|=\nu$ and $|R^B|=\mu$.

This notation is somewhat intertwined with the square bracket partition properties. Namely, letting $\kappa\geq\lambda$ and $\kappa\geq\mu\geq\nu>\omega$, the partition property $\kappa\rightarrow[\mu]^{<\omega}_{\lambda,<\nu}$ is equivalent to the existence of some $\rho<\nu$ such that $(\kappa,\lambda)\twoheadrightarrow(\mu,\rho)$. 

As a result, some large cardinal axioms and partition properties can be described with this notation. In particular:

• $\kappa$ is Rowbottom if and only if $\kappa>\aleph_1$ and for any uncountable $\lambda<\kappa$, $(\kappa,\lambda)\twoheadrightarrow(\kappa,\aleph_0)$. 
• $\kappa$ is Jónsson if and only if for any $\lambda\leq\kappa$, there is some $\nu\leq\kappa$ such that $(\kappa,\lambda)\twoheadrightarrow(\kappa,\nu)$. 

Chang's conjecture is precisely $(\aleph_2,\aleph_1)\twoheadrightarrow(\aleph_1,\aleph_0)$. Chang's conjecture is equivalent to the partition property $\omega_2\rightarrow[\omega_1]_{\aleph_1,<\aleph_1}^{<\omega}$. 

## Variants

There are many stronger variants of Chang's conjecture. Here are a few and their upper bounds for consistency strength (all can be found in ):

• Assuming the consistency of a $\kappa$ which is $\kappa^{++}$-supercompact, it is consistent that there is a proper class of cardinals $\lambda$ such that $(\lambda^{+++},\lambda^{++})\twoheadrightarrow(\lambda^+,\lambda)$.
• Assuming the consistency of a $\kappa$ which is $\kappa^{++}$-supercompact, it is consistent that there is a proper class of cardinals $\kappa$ such that $(\lambda^{+\omega+2},\lambda^{+\omega+1})\twoheadrightarrow(\lambda^+,\lambda)$.
• Assuming the consistency of a cardinal $\kappa$ which is $\kappa^{+\omega+1}$-supercompact, it is consistent that $(\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\aleph_1,\aleph_0)$.
• Assuming the consistency of a huge cardinal, it is consistent that $(\kappa^{++},\kappa^+)\twoheadrightarrow(\mu^+,\mu)$ for every $\kappa$ and $\mu<\kappa^+$.
• It is unknown whether or not it is consistent that $(\aleph_{\omega_1+1},\aleph_{\omega_1})\twoheadrightarrow(\aleph_{\omega+1},\aleph_\omega)$.