8
$\begingroup$

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.

$\endgroup$
13
  • $\begingroup$ 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... $\endgroup$ Sep 6, 2017 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. $\endgroup$ Sep 6, 2017 at 18:11
  • $\begingroup$ @Dan 1. Which result of Jensen? 2. Does inaccessible="strongly inaccessible"? $\endgroup$
    – saf
    Sep 6, 2017 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 $\endgroup$ Sep 6, 2017 at 18:16
  • 1
    $\begingroup$ @Dan Right. That's why we needed ams.org/mathscinet-getitem?mr=3271280 $\endgroup$
    – saf
    Sep 7, 2017 at 7:02

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.