Spanning surfaces in \(3\)-graphs (Q2119367)

Summary: We prove a topological extension of Dirac's theorem suggested by Gowers in 2005 for any connected, closed surface \(\mathscr{S}\), 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 \(\mathscr{S}\) 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.