# Partition relations, Suslin trees and Jónsson cardinals — an interplay of open problems

I’ve previously mentioned the open problem of whether inaccessible Jónssons provably are weakly compact. Here I want to present a few other seemingly completely different open problems in set theory raised by various people and show the interaction between them.

A Jónsson cardinal is one of those cardinals that can be characterised in multiple very different ways. Here are some of them.

Definition. A cardinal $\kappa$ is a Jónsson cardinal if it satisfies any of the following.

1. (Jónsson ’62) Every algebraic structure of size $\kappa$ in a countable language has a proper subalgebra of the same size;
2. (Erdős-Hajnal ’66) $\kappa\to[\kappa]^{<\omega}_\kappa$; i.e. for every $f:[\kappa]^{<\omega}\to\kappa$ there’s an $H\in[\kappa]^\kappa$ such that $f"[H]^{<\omega}\neq\kappa$;
3. (Kleinberg ’79) $(\kappa,\nu)\twoheadrightarrow (\kappa,<\nu)$ for some $\nu<\kappa$; i.e. for every structure $\mathcal M:=(M,\in,A)$ with $|M|=\kappa$ and $|A|=\nu$ there’s an elementary substructure $(H,\in,B)\prec\mathcal M$ such that $|H|=\kappa$ and $|B|<\nu$;
4. (Tryba ’84) For every $\theta>\kappa$ there’s a transitive M and an elementary embedding $j:M\to H_\theta$ with $\text{crit }j<\kappa$ and $j(\kappa)=\kappa$.

I have mentioned that inaccessible Jónsson cardinals share a lot of common properties with weakly compact cardinals. This includes threadability, Mahloness, reflecting stationary sets and there not being any $\kappa$Suslin trees. But the following is open (but conjectured by Welch (’98) to be false).

Question 1. Is every inaccessible Jónsson cardinal provably weakly compact?

Curiously though, the similarities with weakly compacts persists if we weaken the notion of a Jónsson cardinal considerably: say an uncountable cardinal $\kappa$ is weakly Jónsson if $\kappa\to[\kappa]^2_\kappa$. Then whenever $\kappa$ is weakly Jónsson, it holds that

1. (Soare, mentioned in Jensen ’72) There exists no $\kappa$-Suslin tree;
2. (Todorčević ’87) Every stationary $S\subseteq\kappa$ reflects;
3. (Shelah ’94) if $\kappa$ is inaccessible then it’s $\omega$-Mahlo;
4. (Rinot ’14) $\kappa$ is threadable.

These are basically all the same properties that have been shown to hold for inaccessible Jónssons, but the main difference here is that whereas Jónsson cardinals are equiconsistent with Ramsey cardinals, weakly Jónsson cardinals are equiconsistent with weakly compacts. This is basically because $\kappa$ is weakly compact if and only if $\kappa\to[\kappa]^2_2$, so every weakly compact cardinal is weakly Jónsson, and it’s a result of Todorčević (’87) that any threadable cardinal $\kappa\geq\aleph_2$ is weakly compact in L (and he also showed that $\aleph_1$ isn’t weakly Jónsson). We can thus ask the following question.

Question 2. Is every inaccessible weakly Jónsson cardinal provably weakly compact?

A positive answer to Question 2 would then of course also answer Question 1 affirmatively. Rewriting Question 2 in terms of combinatorics we get the following equivalent problem, which is problem 16 in Erdős-Hajnal (’71), that has been open for almost fifty years (I thank Péter Komjáth for pointing this out).

Question 2* (Erdős-Hajnal). For an inaccessible cardinal $\kappa$, does $\kappa\to[\kappa]^2_\kappa$ imply $\kappa\to(\kappa)^2$?

Another related question is the following, posed as Question 8.5 in Shelah (’91) and pointed out by Todd Eisworth.

Question 3 (Shelah). Does $\kappa\to[\kappa]^2_\kappa$ hold for the first $\omega$-Mahlo cardinal $\kappa$?

If Question 3 is consistently false then we have a model in which an uncountable weakly Jónsson isn’t weakly compact (as weakly compacts have many $\omega$-Mahlos below them), falsifying Question 1. Moving on to trees, Mohammad Golshani posed the following.

Question 4 (Golshani). Is it consistent that there are no $\kappa$Suslin trees for some inaccessible but not weakly compact $\kappa$?

As I mentioned above, there are no $\kappa$-Suslin trees for weakly Jónsson cardinals $\kappa$, so a negative answer to Question 1 would then yield a positive answer to Question 4. So summing up,

A negative answer to Shelah’s Question 3 would falsify Question 1, which in turn would falsify Question 2 (and thus also Erdős’ and Hajnal’s Question 2*) and answer Golshani’s Question 4 in the positive.