Spanning surfaces in 3-graphs

Abstract

We prove a topological extension of Dirac's theorem suggested by Gowers in 2005: for any connected, closed surface $\mathcal{S}$, we show that any two-dimensional simplicial complex on $n$ vertices in which each pair of vertices belongs to at least $\frac{n}{3}+o(n)$ facets contains a homeomorph of $\mathcal{S}$ spanning all the vertices. This result is asymptotically sharp, and implies in particular that any 3-uniform hypergraph on $n$ vertices with minimum codegree exceeding $\frac{n}{3}+o(n)$ contains a spanning triangulation of the sphere.