What on Earth do Russian Matryoshka dolls have in common with large cardinal axioms?! Well, the answer lies in Jónsson algebras! Here is how:

As illustrated in the pictures, a Matryoshka set is a self-replicating *container* in which the *mother doll* contains a similar copy of itself. This *sub-doll* also contains a rather smaller similar copy of itself and so on. The dolls in a Matryoshka set could be completely *isomorphic in shape* (often depicted as a chubby Russian girl wearing Sarafan, the beautiful Persian-named traditional Russian dress) or with some *slight differences* in spite of general similarity (e.g. Different species of Taiga).

As the allegory suggests, the *Matryoshka phenomenon* is not uncommon in mathematics. There are (infinite) self-replicating mathematical structures which contain some proper isomorphic copies of themselves or at least have a *fairly similar* (e.g. elementary) proper sub-structures of the same size as the mother structure. Jónsson algebras provide an iconic example of this type.

Definition 1.Let $M$ be a model in a countable language. $M$ is called a Jónsson algebra (model) if it does not have any proper (elementary) submodel of the size $|M|$.

It is easy to see that for an infinite cardinal $\kappa$, the existence of a Jónsson algebra of size $\kappa$ is equivalent to the existence of a Jónsson model of size $\kappa$. By the way, there is a combinatorial large cardinal of the same name:

Definition 2.An uncountable cardinal number $\kappa$ is called Jónsson if for every function $f:[\kappa]^{<\omega}\rightarrow \kappa$ there is a set $H$ of order type $\kappa$ such that for each $n$, $f$ restricted to $n$-element subsets of $H$ omits at least one value in $\kappa$.

According to the following fact, it turns out that the demand for every infinite first-order structure in a countable language to have a proper sub-structure of the same size (i.e. non-existence of Jónsson algebras) has the large cardinal strength of a Jónsson cardinal which is a fairly strong large cardinal assumption incompatible with the Axiom of Constructibility.

Theorem.For an infinite cardinal $\kappa$, the followings are equivalent:(a) $\kappa$ is a Jónsson cardinal.

(b) There are no Jónsson algebras of size $\kappa$. (i.e. Every algebra of size $\kappa$ is a

Matryoshka, namely every algebra of size $\kappa$ contains a proper sub-algebra of the same size.)

The above fact indicates that the occurrence of a *Matryoshka phenomenon* (i.e. having *self-replicating* structures) among certain types of mathematical structures may give rise to some large cardinals. In other words, a structure needs to be *so large* in order to be able to support a self-replicating Matryoshka-like structure. In this sense, Jónsson cardinals have a characterization in terms of a Matryoshka phenomenon for certain type of mathematical structures, namely algebras.

Here, the natural question to ask is whether there are other examples of a Matryoshka characterization for large cardinal axioms. In other words, what other Matryoshka phenomenona for mathematical structures do have some large cardinal strength? Precisely:

Question 1.What are some other examples of large cardinal characterizations of the following form?The followings are equivalent/equiconsistent:

(a) $\kappa$ is large.

(b) Any structure of type X and size $\kappa$ has a proper (isomorphic, elementary, "similar", etc.) sub-structure of size $\kappa$. (i.e. There is a Matryoshka phenomenon among mathematical structures of type X).

Note that we are not limiting ourselves to the first order structures or structures in countable language. The question is considering the possible existence of such a characterization for large cardinal axioms *anywhere* in mathematics where a theory around certain type of "mathematical structures" (whatever it means depending on the context, say functional space or a manifold) evolves.

Question 2.How much does the consistency strength of $\kappa$ increase if we expand our demand in the existing Matroshka chracaterization of Jónsson cardinals to containisomorphicproper sub-algebras? Will we hit the inconsistency barrier? Precisely, what is the consistency strength of the following statement? Is it consistent at all?"Every algebra of size $\kappa$ contains a proper sub-algebra of the same size which is

isomorphicto the original structure."