Spanning surfaces in 3-graphs
Agelos Georgakopoulos, John Haslegrave, Richard Montgomery, Bhargav P. Narayanan · Journal of the European Mathematical Society · 2021
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 n3 + 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 n3+o(n) contains a spanning triangulation of the sphere.