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Recall that the square partition relation $\kappa\to[\lambda]^k_\eta$ holds iff for every $f:[\kappa]^k\to\eta$ there exists $H\in[\kappa]^\lambda$ such that $f"[H]^k\neq\eta$. I.e. said in words, everytime we colour the $k$-element subsets of $\kappa$ in $\eta$ many colours, we can find a subset $H\subseteq\kappa$ of size $\lambda$ such that we omit a colour when colouring $k$-element subsets of $H$.

Note that for $\kappa>\omega$, $\kappa\to[\kappa]^2_2$ holds iff $\kappa$ is weakly compact, and it's not too hard to see that $\kappa\to[\kappa]^2_2$ implies $\kappa\to[\kappa]^2_\kappa$. Furthermore, it's a result of Rinot ('14) that every regular uncountable cardinal $\kappa$ satisfying $\kappa\to[\kappa]^2_\kappa$ is threadable, which by a result of Todorcevic ('87) implies that such a cardinal is weakly compact in $L$, making the two notions equiconsistent. This leads me to my question.

Question. Can we separate inaccessible cardinals $\kappa$ satisfying $\kappa\to[\kappa]^2_\kappa$ from the weakly compacts? By this I mean showing that $\textsf{ZFC}+\exists\text{weakly compact}$ can't prove that every inaccessible $\kappa$ satisfying $\kappa\to[\kappa]^2_\kappa$ is weakly compact.

The standard way to do this is to use Kunen's method of starting with a weakly compact cardinal, adding a $\kappa$-Souslin tree to kill the weak compactness, resurrect the weak compactness in a further forcing extension and show that $\kappa$ has the desired property in the intermediate extension. But the presence of such a tree either makes $\kappa$ singular or implies that $\kappa\not\to[\kappa]^2_\kappa$ (this is mentioned on slide 12 of Rinot's ESTC '17 talk), so that doesn't work in this case.

EDIT: This edit is based on the comments below. Replaced Todorcevic' result that regular uncountable cardinals $\kappa$ satisfying $\kappa\to[\kappa]_\kappa^2$ reflects stationary sets to Rinot's result that such cardinals are threadable. Both are true, but we need threadability to show the equiconsistency.

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  • $\begingroup$ This is related to Question 8.5 in Shelah's "On what I do not understand (and have something to say) Part I" [Sh:666], where he asks specifically if $\lambda\nrightarrow [\lambda]^2_\lambda$ holds for the least $\omega$-Mahlo cardinal. If $\lambda$ is (weakly) inaccessible not $\omega$-Mahlo, then you get some strong colorings of pairs, but this is buried in Chapter 4 of Cardinal Arithmetic and pretty hairy... $\endgroup$ – Todd Eisworth Sep 6 '17 at 17:16
  • $\begingroup$ This is problem 16 in the problem list of Erdos and Hajnal: Unsolved problems in set theory renyi.mta.hu/~p_erdos/1971-28.pdf. I started writing up a paper on those problems, will be happy to send it to you when finished. $\endgroup$ – Péter Komjáth Sep 6 '17 at 18:11
  • $\begingroup$ @Dan 1. Which result of Jensen? 2. Does inaccessible="strongly inaccessible"? $\endgroup$ – saf Sep 6 '17 at 18:14
  • $\begingroup$ @saf 1. I was thinking of the result that every regular cardinal reflecting stationary sets is weakly compact in $L$. I believe it's due to Jensen, but correct me if I'm wrong. 2. Yes $\endgroup$ – Dan Saattrup Nielsen Sep 6 '17 at 18:16
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    $\begingroup$ @Dan Right. That's why we needed ams.org/mathscinet-getitem?mr=3271280 $\endgroup$ – saf Sep 7 '17 at 7:02

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