One-to-one Borel selection theorems

For X = [0, 1] we obtain new theorems stating that a Borel set in X-2 with large sets of large vertical and large horizontal sections admits a one-to- one Borel selection with large domain and large range. Largeness is meant m ainly in measure or category sense. Our proofs combine a result of Graf and Mauldin with a modified result of Sarbadhikari.