Abstract: Free boundaries of solutions to the one-phase problem can have rich geometry. Such richness can be reduced by imposing the graphical condition. In this talk, we show that if homogeneous minimizers are trivial in dimension k, then graphical solutions are trivial in dimension k+1. This works for both the classical one-phase problem as well as its thin counterpart. This talk is based on a joint work with Max Engelstein (Minnesota) and Xavier Fernandez-Real (EPFL).