The Turán Number of Surfaces

We show that there is a constant $c$ such that any 3-uniform hypergraph $\mathcal{H}$ with $n$ vertices and at least $c{n}^{5\mathrm{/}2}$ edges contains a triangulation of the real projective plane as a subgraph. This resolves a conjecture of Kupavskii, Polyanskii, Tomon, and Zakharov. Furthermore, our work, combined with prior results, asymptotically determines the Turán number of all surfaces.