Proof of Caldararu's conjecture. An appendix to a paper by Yoshioka
  We show that any orientation preserving Hodge isometry between the Hodge structures of two K3 surfaces X and X' twisted by Brauer classes $\alpha$ resp. $\alpha'$ can be lifted to a Fourier-Mukai equivalence between the derived categories of $\alpha$ resp. $\alpha'$-twisted sheaves. 
Daniel Huybrechts and Paolo Stellari