What is the strongest known natural large cardinal axiom compatible with $V = L$ (strongest in the sense that it implies all known "small" large cardinal axioms, where a large cardinal axiom is said to be "small" if it doesn't imply $V \neq L$)?


One candidate might be "there is an $\alpha$-Erdos cardinal for every countable ordinal $\alpha$."


Using the ``instrumentalist dodge'' of Steel/Hamkins, it seems that we can "cheat" to obtain stronger large cardinal axioms within the confines of $\operatorname{ZF}+(V = L)$, such as "there is a transitive model of the theory ZFC+"$0^\sharp$ exists."" I will leave whether or not such axioms are "natural" for a matter of debate here. 


Disclaimer: I'm an algebraist and number theorist but am currently trying to grasp the big picture in set theory, large cardinals, inner model theory, and the like.


Related MO posts:


[Is there a large-cardinal completeness theorem for $L$?](https://mathoverflow.net/questions/168938/)


[If $0^{\sharp}$ exists, then every uncountable cardinal in $V$ is as large as it can be in $L$.](https://mathoverflow.net/questions/61764/)


[Erdős cardinals and $0^\sharp$](https://mathoverflow.net/questions/183070/)



Strongest large cardinal axiom compatible with $V = L$?
Maybe a comment:


In the paper ``[A large cardinal in the constructible universe](http://matwbn.icm.edu.pl/ksiazki/fm/fm69/fm6916.pdf)'' Silver shows that if $\kappa\to (\alpha)^{<\aleph\_0}$ for all countable $\alpha,$ then the same is true for $\kappa$ in $L$.


On the other hand, by a result of Rowbottom, $\kappa \to (\omega\_1)^{<\aleph\_0} $ contradicts $V=L.$


Silver concludes with the following:



> 
> It does not seem extravagant, then, to assert that, for all practical purposes, $\kappa \to (\omega)^{<\aleph\_0}$ is the strongest strong axiom of infinity know to be consistent with $V=L,$ and therein lies its chief interest. 
> 
> 
> 


