AbstractWe prove a conjecture of D. Oberlin on the dimension of unions of lines in . If is an integer, , and L is a set of lines in with Hausdorff dimension at least , then the union of the lines in L has Hausdorff dimension at least . Our proof combines a refined version of the multilinear Kakeya theorem by Carbery and Valdimarsson with the multilinear → linear argument of Bourgain and Guth.