TITLE: Perfect set property for projective hierarchy
QUESTION [9 upvotes]: Is there any paper discussing the consistency strength (or possible equivalents, maybe large cardinals) of just assuming the perfect set property for certain levels of the projective hierarchy?

REPLY [11 votes]: The classical argument Andres mentions can be taken one step further to show that if every uncountable $\Pi_1^1$ set contains a perfect subset then $\aleph_1$ is inaccessible in $L$.
This follows from the same fact for $\Sigma^1_2$-sets since, by the uniformization theorem,
every $\Sigma^1_2$-set is the injective projection of a $\Pi^1_1$-set.