# Historical question about the $\aleph_2$-Souslin hypothesis

For an uncountable regular cardinals $$\kappa,$$ let $$\kappa$$-Souslin hypothesis, denoted $$SH(\kappa)$$ be the assertion that there are no $$\kappa$$-Souslin trees.

By a result of Jensen, $$GCH+SH(\aleph_1)$$ is consistent. However, the problem of the consistency of $$GCH+SH(\aleph_2)$$ is still open.

Question. $$(a)$$ Where the above question first appeared?

$$(b)$$ Who first asked the question (possibly not in a published paper or book) and when?

Any references are welcome.

First, in case your question suggests that you managed to prove the consistency of $GCH+SH(\omega_2)$, then let me congratulate you wholeheartedly! Second, to put things in context, let us recall that an $\omega_2$-Souslin tree is a non-special $\omega_2$-Aronszajn tree which is (obviously) an $\omega_2$-Aronszajn tree.

Back to your question. By the time that Jensen proved the consistency of $GCH+SH(\omega_1)$, all of the following were already known:

(1) If $CH$ holds, then there exists a special $\omega_2$-Aronszajn tree (Specker, 1949).

(2) If $V=L$, then $GCH$ holds and there is an $\omega_2$-Souslin tree (Jensen, 1972).

(3) It is equiconsistent with the existence of a weakly compact cardinal that there are no $\omega_2$-Aronszajn trees (Mitchell [and Silver], 1972).

Anyone who was aware of the last three results was basically led to the question of whether $GCH$ (or even just $CH$) is consistent with $SH(\omega_2)$. I do not think this automatic'' question is attributed to anyone.

As far as I can tell, the first paper to address this problem was Gregory's (1976). Recall that Jensen's theorem concerning $V=L$ actually gave two sufficient conditions for the existence of an $\omega_2$-Souslin tree: (1) $CH+\diamondsuit(S^2_1)$, and (2) $\square_{\omega_1}+\diamondsuit(S^2_0)$. In the above-mentioned paper, Gregory proved that $\diamondsuit^+(S^2_0)$ follows from $GCH$ (this should be compared with a 1980 theorem of Shelah who proved that $\diamondsuit(S^2_1)$ does not follow from $GCH$). In the same paper, Gregory furthermore proved that $GCH$ and the existence of a non-reflecting stationary subset of $S^2_0$ (a consequence of $\square_{\omega_1}$) already yields such a tree. Gregory's result and the surrounding questions were soon after echoed in a survey paper by Kanamori and Magidor (1978).

Around the same time, Laver proved that the consistency of a measurable cardinal yields the consistency of $CH+SH(\omega_2)$. I recently learned that in a visit to Berkeley, Laver said that he is looking to reduce the hypothesis into a weakly compact cardinal, since this would gave an equiconsistency''. In a phone conversation I had with Shelah this very week, he told me the following. When Laver proved his consistency result, Mati Rubin (who got his Ph.D. with Shelah) was in Colorado. Rubin informed Shelah of the proof, and Shelah managed to reduce the measurable cardinal hypothesis into a weakly compact. After a wait period, Shelah airmailed his proof to Laver, who wrote down a joint paper (1981). That is, the paper was physically written by Laver.

The last paragraph of the Laver-Shelah paper (1981) reiterates the fact it is open whether $GCH+SH(\omega_2)$ is consistent. This paragraph refers to Gregory's 1976 paper that shows that $GCH+SH(\omega_2)$ implies the consistency of a Mahlo cardinal, and to an upcoming paper of Shelah and Stanely (that appeared in 1982) showing that $CH+SH(\omega_2)$ implies the consistency of an inaccessible cardinal. A proof (or just a claim) that the Laver-Shelah theorem gives an equiconsistency result never appeared in print.

Shelah also told me that for many years Gregory was claiming that he knows how to get the consistency of $GCH+SH(\omega_2)$. Again, such a paper never appeared.

The last page of the Laver-Shelah paper points out that the same technique of their proof shows that the consistency of a weakly compact cardinal yields the consistency of $CH$ together with all $\omega_2$-Aronszajn trees are special. This should be compared with a theorem of Shelah-Stanley (1988) and Todorcevic (1989) that if $\omega_2$ is not weakly compact in $L$, then there is a non-special $\omega_2$-Aronszajn tree.

Moving forward to 1992, Kojman and Shelah published a paper that improves Gregory's 1976 theorem to show that if $GCH$ holds and there exists a non-reflecting stationary subset of $S^2_0$, then there exists an $\omega_2$-Souslin tree which is moreover countably complete. Their proof replaced Jensen's $\diamondsuit(S^2_1)$ hypothesis with some weak form of club-guessing at $S^2_1$. In their paper, Kojman and Shelah mention that the referee of the paper claimed that Gregory knew how to get a countably complete $\omega_2$-Souslin tree from $GCH+\square_{\omega_1}$. Personally, I doubt that Gregory anticipated club-guessing, and have no idea how else could his proof go. In any case, Gregory's proof was never published.

Coming back to your question: the fact that $GCH+SH(\omega_2)$ is open appears in the opening paragraph of the Kojman-Shelah paper, and the reader is referred to Gregory (1976), Laver-Shelah (1981), and Shelah-Stanley (1982). The paper is concluded by reiterating the problem of whether $GCH+SH(\omega_2)$ requires the consistency of a weakly compact. According to Shelah (and to the literature), these questions remained unanswered.

In a paper from 2007, Konig, Larson, and Yoshinobu introduced a generalized club-guessing principle at $S^2_1$ and proved that the conjunction of their principle with $GCH$ yields a (countably closed) $\omega_2$-Souslin tree. In response, in a paper from 2011, I proved that their principle does not follow from $GCH$, and hence their approach cannot shed any light on the problem of $GCH+SH(\omega_2)$.

In a paper from from 1974, Erdos and Hajnal asked whether a poset of size $\aleph_2$ that does not contain a copy of $\omega_2$ or of $-\omega_2$, must contain a suborder of size $\aleph_1$ that does not contain a copy of $\omega_1$ or of $-\omega_1$. In paper from 1981, Todorcevic proved that the consistency of a weakly compact cardinal yields the consistency of $GCH$ together with the assertion that any $\omega_2$-Aronszajn tree contains an $\omega_1$-Aronszajn subtree. In paper from 2017, I proved that if $GCH$ holds and $\omega_2$ is not weakly compact in $L$, then there exists an $\omega_2$-Souslin tree with no $\omega_1$-Aronszajn subtrees. The proof is motivated by the Ostaszewski square'' principle that I introduced in a paper from 2014. The latter asserts the existence of a $\square_{\omega_1}$-sequence with some built-in (strong form of) club-guessing. Roughly speaking, the new ingredient of the 2017 paper was to show that $\square_{\omega_1}$ may be relaxed to $\square(\omega_2)$. In the same paper, I also pushed further the Kojman-Shelah argument and showed that $GCH+\square(\omega_2)$ entails the existence of an $\omega_2$-Souslin tree which is countably complete.

There are further exciting news about $\omega_2$-Aronszajn trees obtained recently by Krueger, and by Lambie-Hanson and Lucke, but these do not directly address the problem of $GCH+SH(\omega_2)$.