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We prove some epsilon regularity results for n-dimensional minimal two-valuedLipschitz graphs. The main theorems imply uniqueness of tangent cones andregularity of the singular set in a neighbourhood of any point at which atleast one tangent cone is equal to a pair of transversely intersectingmultiplicity one n-dimensional planes, and in a neighbourhood of any point atwhich at which at least one tangent cone is equal to a union of four distinctmultiplicity one n-dimensional half-planes that meet along an (n-1) -dimensional axis. The key ingredient is a new Excess Improvement Lemma obtainedvia a blow-up method (inspired by the work of L. Simon on the singularities of`multiplicity one' classes of minimal submanifolds) and which can be iteratedunconditionally. We also show that any tangent cone to an n-dimensional minimaltwo-valued Lipschitz graph that is translation invariant along an (n-1) or(n-2)- dimensional subspace is indeed a cone of one of the two aforementionedforms, which yields a global decomposition result for the singular set

Transverse Singularities of Minimal Two-Valued Graphs in Arbitrary Codimension