The $ \mathbf{\Sigma}^1_2$ counterparts to statements that are   equivalent to the Continuum Hypothesis
  We consider natural $\Sigma^1_2$ definable analogues of many of the classical statements that have been shown to be equivalent to CH. It is shown that these $\Sigma^1_2$ analogues are equivalent to that all reals are constructible. We also prove two partition relations for $\Sigma^1_2$ colourings which hold precisely when there is a non-constructible real. 
Asger Tornquist and William Weiss
math.LO