Free boundary partial regularity in the thin obstacle problem

For the thin obstacle problem in $\mathbb{R}^n$, $n\geq 2$, we prove that at all free boundary points, with the exception of a $(n-3)$-dimensional set, the solution differs from its blow-up by higher order corrections. This expansion entails a $C^{1,1}$-type free boundary regularity result, up to a codimension 3 set.