  We prove a Bernstein-type theorem for two-valued minimal graphs in the
four-dimensional Euclidean space $\mathbf{R}^4$. This states that two-valued
functions defined on the entire $\mathbf{R}^3$, and whose graph is a minimal
surface, must necessarily be linear. This is a two-valued analogue of the
classical Bernstein theorem, which asserts that in dimensions up to $n+1 \leq
8$, an entire single-valued minimal graph is linear. The main contrast with the
single-valued theory is the presence of a large set of singularities in the
graphs of two-valued functions. Indeed two-valued minimal graphs are neither
area-minimising, nor is the regularity theory of elliptic PDE directly
available in this setting. We obtain structure results for the blowdown cones
of two-valued minimal graphs, valid in dimension $n+1 \leq 7$, proving in
particular that they are smoothly immersed away from an $(n-2)$-rectifiable set
that includes its branch points. In dimension four we go further, and
completely classify the possible blowdown cones using a combinatorial argument.
We show that they must be a union of two three-dimensional planes: this is the
key to the proof of the Bernstein theorem.
