Extremal set theory tries to answer questions about the maximal or minimal size of subsets of some universal set, while respecting certain imposed restrictions. In this talk we will discuss one such example, which is known as the Frankl-Wilson theorem. This theorem turns out to have applications in geometry, providing lower bounds on both the chromatic number of Euclidean space and the number of parts one needs to subdivide a bounded region in Euclidean space into smaller regions.