  Two-valued sets are local sets of the two-dimensional Gaussian free field
(GFF) that can be thought of as representing all points of the domain that may
be connected to the boundary by a curve on which the GFF takes values only in
[-a,b]. Two-valued sets exist whenever $a+b\geq 2\lambda$, where $\lambda$
depends explicitly on the normalization of the GFF. We prove that the almost
sure Hausdorff dimension of the two-valued set $A_{-a,b}$ equals
$d=2-2\lambda^2/(a+b)^2$. For the two-point estimate, we use the real part of a
"vertex field" built from the purely imaginary Gaussian multiplicative chaos.
We also construct a non-trivial $d$-dimensional measure supported on $A_{-a,b}$
and discuss its relation with the $d$-dimensional conformal Minkowski content
for $A_{-a,b}$.
