# Journal of the European Mathematical Society

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**Published online first: 2021-08-24**

**DOI: 10.4171/JEMS/1101**

Spanning surfaces in 3-graphs

Agelos Georgakopoulos^{[1]}, John Haslegrave

^{[2]}, Richard Montgomery

^{[3]}and Bhargav Narayanan

^{[4]}(1) University of Warwick, Coventry, UK

(2) University of Warwick, Coventry, UK

(3) University of Birmingham, UK

(4) Rutgers University, Piscataway, USA

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.

*Keywords: *Extremal simplicial topology, spanning structures in hypergraphs, Dirac’s theorem, triangulated surfaces

Georgakopoulos Agelos, Haslegrave John, Montgomery Richard, Narayanan Bhargav: Spanning surfaces in 3-graphs. *J. Eur. Math. Soc.* Electronically published on August 24, 2021. doi: 10.4171/JEMS/1101 (to appear in print)