I will discuss lower bounds on the dimension of a thin set, \(E\), ensuring that the set of all \(k\)-point configurations in \(E\) of a given type has nonempty interior. Here, `configurations’ are determined by (possibly vector-valued) functions of \(k\) points in \(E\), and include congruence classes of standard geometric figures (line segments, triangles, etc.), but are considerably more general. After giving some background, I will discuss joint work with Alex Iosevich and Krystal Taylor on results of this type using estimates for multilinear Radon transforms obtained by what we call partition optimization, and compute a few examples.