The Extremal Number of Surfaces

Fix a set V of finitely many points and consider those triangles whose vertices are in V. Consider a geometric space X constructed by taking the union of some of these triangles. How many triangles can X have if no subset of these triangles triangulates a sphere? This question (which is secretly about hypergraphs) was answered by Sós, Erdős, and Brown. We present their argument and discuss more recent extensions to surfaces and other topological spaces.