Consistency of many Erdos cardinals Does anyone have a reference for the consistency strength of saying that the Erdos cardinal $\kappa(\alpha)$ exists for all $\alpha$?  Or of various weakenings about how far such cardinals extend into $\text{Ord}$?
This comes up while looking at the consistency strength of sharps of sets of ordinals, so any reference on that front would also be welcome.
 A: I agree with the last sentence of Bill Mitchell's comment. But here is something closer than the cardinals mentioned. In [1] the notion of "almost Ramsey" cardinal was coined. Such a cardinal $\kappa$ is required to be $\alpha$-Erdos for all $\alpha<\kappa$. Then $V_\kappa$ is a model of what you are after (but in fact this is still not exact as there are many other $\gamma <\kappa$ for which this is true too. Almost Ramseys also get an outing in [2].
If you are interested in sharps alone, then, eg just for sharps for reals, $\kappa \rightarrow (\omega_1)^{<\omega}_{2}$ is already overkill: one really just needs for any function $f$ homogeneous sets of arbitrarily large but countable length, all of which have the same "type".  This has been investigated closely in [3].  Similar considerations would hold for sharps of other sets of ordinals.  The least inner model in which every set has a sharp, $L^\#$ say, is too thin to contain any Erdos cardinals (other than trivially $\kappa(\delta)$ for $\delta< \omega_1^{L^\#}$).  
[1] J. Vickers & P.Welch "Elementary Embeddings of an inner model into the Universe" JSL vol 66, 2001.
[2] A. Apter & P. Koepke "Making All cardinals almost Ramsey" Archive for Math. Logic, vol 47, 2008.
[3] J. Baumgartner & F.Galvin "Generalized Erdos cardinals and $0^\#$", Ann. of Math. Logic, vol. 15 , 1978.
