During the 1970’s Erdos, Graham at al. initiated a study of geometric point configurations which cannot be destroyed by finite partitions of high dimensional Euclidean spaces. They have shown that such configurations must be spherical and in 1994 Graham conjectured that this condition is also sufficient. While these problems were initially studied by purely combinatorial means a surprising density analogue has been shown for simplexes by Bourgain, using Fourier analysis. We will present some recent results toward this conjecture both in the Euclidean setting and in vector spaces over finite fields, and discuss an approach utilizing some modern tools of additive combinatorics, also referred to as “higher order Fourier analysis”.