What large cardinal axiom, if any, implies that there are unboundedly many Ramsey cardinals below the large cardinal?

That is, is there a large cardinal axiom, such that if $\kappa$ is that large cardinal there are unboundedly many Ramsey cardinals $< \kappa$.


(I removed the past version of the answer, since I read "unboundedly many Ramsey cardinals" as "proper class of Ramsey cardinals", which is perhaps the more natural reading for a set theorist.)

If $\kappa$ is measurable then it is the limit of Ramsey cardinals, and in fact it is a measure $1$ limit of Ramsey cardinals. To see that, note that (1) measurable is Ramsey; and (2) if $M$ is an inner model that agrees with $V$ up to $V_{\alpha+1}$, then $M$ agrees with $V$ on whether or not $\alpha$ is a Ramsey cardinal, since being Ramsey is a 2nd order property.

Combine the two facts, and we see that if $j\colon V\to M$ witnessing that $\kappa$ is measurable, then $V_{\kappa+1}\subseteq M$, so $\kappa$ is Ramsey in $M$. But then the set of Ramsey cardinals below $\kappa$ has measure $1$.


Of course, we can just look at "worldly limit of Ramsey cardinals" or "inaccessible limit of Ramsey cardinals", etc. Much weaker axioms.

  • $\begingroup$ Regarding the last paragraph: Woodin cardinals are not measurable, but are an inaccessible limit of measurable cardinals. $\endgroup$ Apr 25 '20 at 0:53
  • 1
    $\begingroup$ (And, again, congratulations!) 🥳 $\endgroup$ Apr 25 '20 at 0:54
  • $\begingroup$ @AndrésE.Caicedo: Thanks! :) $\endgroup$
    – Asaf Karagila
    Apr 25 '20 at 7:18

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