# Spanning surfaces in 3-graphs

### Agelos Georgakopoulos

University of Warwick, Coventry, UK### John Haslegrave

University of Warwick, Coventry, UK### Richard Montgomery

University of Birmingham, UK### Bhargav Narayanan

Rutgers University, Piscataway, USA

## Abstract

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.

## Cite this article

Agelos Georgakopoulos, John Haslegrave, Richard Montgomery, Bhargav Narayanan, Spanning surfaces in 3-graphs. J. Eur. Math. Soc. 24 (2022), no. 1, pp. 303–339

DOI 10.4171/JEMS/1101