It is well-known that if the Erdős cardinal $\kappa(\omega\_1)$ exists, then $0^\sharp$ exists, but what if $\kappa(\lambda)$ exists for a limit ordinal $\omega\_1^L\leq \lambda<\omega\_1$? Does this still impliy that $0^\sharp$ exists?


Or, alternatively: it is known that the existence of $\kappa(\lambda)$ for all countable $\lambda$ is consistent with $0^\sharp$ does not exist (with $V=L$, in fact), but, does it imply that $\omega\_1^L=\omega\_1$?



Erdős cardinals and $0^\sharp$
Similarly to Asaf's comment, let $\kappa = \kappa(\omega\_1^L)$ (we assume that it exists). Then in $L\_\kappa$, there is $\alpha$-Erdős cardinal for every $L$-countable $\alpha$ (since being $\alpha$-Erdős cardinal is downward absolute between transitive models for countable ordinals $\alpha$). So $L\_\kappa$ is a model for $\forall \lambda < \omega\_1,\, \kappa(\lambda)$ exists and $\omega\_1 = \omega\_1^L$. 


It is interesting to note that even when $\omega\_1$ is larger then $\omega\_1^L$, still the existence of $\lambda$-Erdős cardinal for every $\lambda < \omega\_1$ does not imply the existence of $0^{\#}$. For example, the existence of $\kappa(\omega\_1^L)$-Erdős cardinals has only mild effect on the cardinals of $L$: Although it does imply that $\omega\_1^L < \omega\_1$ (otherwise $0^{\#}$ exists, and then $\omega\_1^L < \omega\_1$), it is consistent with $\omega\_1 = \omega\_2^L$. The reason is the if $V \models \kappa(\omega\_1^L)$ exists, and $\omega\_2^L$ is countable, then one can construct in $V$ a $L$-generic filter for $Col(\omega, \omega\_1^L)$, $G$, and since $L[G] \subseteq V$ and both agree that $\omega\_1^L$ is countable, the fact that $\kappa(\omega\_1^L)$ is $\omega\_1$-Erdős in $V$, implies that it is also $\omega\_1^L$-Erdős in $L[G]$, but $\omega\_1^{L[G]} = \omega\_2^L$. 


The same argument shows that if in $V$, for every ordinal $\lambda < \omega\_2^L$ we had a $\lambda$-Erdős cardinal, then there is a model in which $\omega\_1 = \omega\_2^L$, and for every countable ordinal $\alpha$ there is a $\alpha$-Erdős cardinal. 


