One of the magnificent theorems of $\sf ZFC$ is that there exists an Aronszajn tree on $\omega_1$. Namely, a tree of height $\omega_1$ in which every level is countable, but no branch is cofinal.

On the other hand, assuming the consistency of a weakly compact, $\sf ZFC$ does not prove that every tree on $\omega_2$—or any successor of an uncountable regular cardinal—which has levels of size less than $\aleph_2$ has a cofinal branch.

This raises a very natural question:

Is there some natural combinatorial property which holds or fails at $\omega_1$ if and only if $\omega_1$ is weakly compact in $L$?

(We can replace "weakly compact in $L$" by "There is a weakly compact cardinal in $L$" if that holds somehow; but usually we expect the cardinal with the failure to be that weakly compact.)

Both in a deleted comment and a deleted answer the Shelah–Harrington theorem was suggested that showed that under $\sf MA+\lnot CH$, $\omega_1$ is weakly compact if and only if every $\mathbf\Sigma^1_3$ is measurable if and only if ever $\mathbf\Delta^1_3$ set has the Baire property.

These are not quite combinatorial characteristics of $\omega_1$, but rather [somewhat-]topological characteristics of the continuum, which is not even $\omega_1$ here. Moreover, if anything is to be quoted from the said paper, it should be the theorem stating that under $\sf MA+\lnot CH$, $\omega_1$ is weakly compact in $L$ if it is inaccessible to reals.

A weakening of the original question could be, suppose $\kappa$ is weakly compact, and we force with $\operatorname{Col}(\omega,<\kappa)$. What combinatorial properties of $\kappa$ survive the collapse, and whose consistency at $\omega_1$ implies weak compactness in $L$?

  • $\begingroup$ That is what Mohammad wrote in his now-deleted answer. But I agree that this is not exactly what I was looking for. This is more of a combinatorial property of the continuum, which is definitely not $\omega_1$. But it is an interesting fact, so thank you both for bringing this up. $\endgroup$ – Asaf Karagila Nov 23 '16 at 22:40
  • $\begingroup$ (Also, if anything, then the result that MA implies that $\omega_1$ is either computable from a real or weakly compact in L is far more relevant here; that's Theorem C(i) in the linked paper.) $\endgroup$ – Asaf Karagila Nov 23 '16 at 22:44
  • $\begingroup$ Yes, I made an edit which was not correct, and deleted it now since you know about it and I was getting into a muddle about it. Perhaps you could mention in your question that such an answer doesn't count? $\endgroup$ – tci Nov 23 '16 at 22:46
  • $\begingroup$ I'll edit this tomorrow. I'll sleep on this and see if the morning will bring some better insights as to what are "natural properties" or what aren't. $\endgroup$ – Asaf Karagila Nov 23 '16 at 22:51
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    $\begingroup$ A characterization is given in the paper "Gitik, M.; Magidor, M.; Woodin, H. Two weak consequences of 0♯. J. Symbolic Logic 50 (1985), no. 3, 597–603." but I don't know if it is interesting to you or not. The statement is "There is a club $C$ of $\aleph_1$ consisting of L-inaccessible cardinals such that for each $\alpha$ a limit point of $C, C \cap \alpha$ is almost contained in every closed unbounded subset of $\alpha$ in L". $\endgroup$ – Mohammad Golshani Nov 24 '16 at 4:48

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