Jónsson cardinal

Jónsson cardinals are named after Bjarni Jónsson, a student of Tarski working in universal algebra. In 1962, he asked whether or or not every algebra of cardinality $\kappa$ has a proper subalgebra of the same cardinality. The cardinals $\kappa$ that satisfy this property are now called Jónsson cardinals.

An algebra of cardinality $\kappa$ is simply a set $A$ of cardinality $\kappa$ and finitely many functions (each with finitely many inputs) $f_0,f_1...f_n$ for which $A$ is closed under every such function. A subalgebra of that algebra is a set $B\subseteq A$ which $B$ is closed under each $f_k$ for $k\leq n$.

Equivalent Definitions

There are several equivalent definitions of Jónsson cardinals.

Partition Property

A cardinal $\kappa$ is Jónsson iff the partition property $\kappa\rightarrow [\kappa]_\kappa^{<\omega}$ holds, i.e. that given any function $f:[\kappa]^{<\omega}\to\kappa$ we can find a subset $H\subseteq\kappa$ of order type $\kappa$ such that $f``[H]^n\neq\kappa$ for every $n<\omega$. [1]

Substructure Characterization

A cardinal $\kappa$ is Jónsson iff given any $A$ there exists an elementary substructure $\langle X,\in, X\cap A\rangle\prec\langle V_\kappa,\in,A\rangle$ with $|X|=\kappa$ and $X\cap\kappa\neq\kappa$.

A cardinal $\kappa$ is Jónsson iff any structure with universe of cardinality $\kappa$ has a proper elementary substructure with universe also having cardinality $\kappa$. [1]

Embedding Characterization

A cardinal $\kappa$ is Jónsson iff for every $\theta>\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to H_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit }j<\kappa$, iff for every $\theta>\kappa$ there exists a transitive set $M$ with $\kappa\in M$ and an elementary embedding $j:M\to V_\theta$ such that $j(\kappa)=\kappa$ and $\text{crit} j<\kappa$.

Interestingly, if one such $\theta>\kappa$ has this property, then every $\theta>\kappa$ has this property as well.

Abstract Algebra Characterization

In terms of abstract algebra, $\kappa$ is Jónsson iff any algebra $A$ of size $\kappa$ has a proper subalgebra of $A$ of size $\kappa$.

Properties

All the following facts can be found in [1]:

$\aleph_0$ is not Jónsson.
If $\kappa$ isn't Jónsson then neither is $\kappa^+$.
If $2^\kappa=\kappa^+$ then $\kappa^+$ isn't Jónsson.
If $\kappa$ is regular then $\kappa^+$ isn't Jónsson (therefore $\kappa^{++}$ is never Jónsson, and if $\kappa$ is weakly inaccessible then $\kappa^+$ is never Jónsson).
A singular limit of measurables is Jónsson.
The least Jónsson is either weakly inaccessible or has cofinality $\omega$.
$\aleph_{\omega+1}$ is not Jónsson.

It is still an open question as to whether or not there is some known large cardinal axiom that implies the consistency of $\aleph_\omega$ being Jónsson.

Relations to other large cardinal notions

Jónsson cardinals have a lot of consistency strength:

Jónsson cardinals are equiconsistent with Ramsey cardinals. [2]
The existence of a Jónsson cardinal $\kappa$ implies the existence of $x^\sharp$ for every $x\in V_\kappa$ (and therefore for every real number $x$, because $\kappa$ is uncountable).

But in terms of size, they're (ostensibly) quite small:

A Jónsson cardinal need not be regular (assuming the consistency of a measurable cardinal).
Every Ramsey cardinal is inaccessible and Jónsson. [3]
Every weakly inaccessible Jónsson is weakly hyper-Mahlo. [4]

It's an open question whether or not every inaccessible Jónsson cardinal is weakly compact.

Jónsson successors of singulars

As mentioned above, $\aleph_{\omega+1}$ is not Jónsson (this is due to Shelah). The question is then if it's possible for any successor of a singular cardinal to be Jónsson. Here is a (non-exhaustive) list of things known:

If $0\neq\gamma<|\eta|$ then $\aleph_{\eta+\gamma+1}$ is not Jónsson. [5]
If there exists a Jónsson successor of a singular cardinal then $0^\dagger$ exists. [6]