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 \kappaSuslin 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 \kappaSuslin 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.