We prove a topological extension of Dirac's theorem suggested by Gowers in 2005: for any connected, closed surface , we show that any two-dimensional simplicial complex on vertices in which each pair of vertices belongs to at least facets contains a homeomorph of spanning all the vertices. This result is asymptotically sharp, and implies in particular that any 3-uniform hypergraph on vertices with minimum codegree exceeding contains a spanning triangulation of the sphere.
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Agelos Georgakopoulos, John Haslegrave, Richard Montgomery, Bhargav Narayanan, Spanning surfaces in 3-graphs. J. Eur. Math. Soc. 24 (2022), no. 1, pp. 303–339DOI 10.4171/JEMS/1101