JournalsjemsVol. 24, No. 1pp. 303–339

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
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Abstract

We prove a topological extension of Dirac's theorem suggested by Gowers in 2005: for any connected, closed surface S\mathscr{S}, we show that any two-dimensional simplicial complex on nn vertices in which each pair of vertices belongs to at least n3+o(n)\frac n3 + o(n) facets contains a homeomorph of S\mathscr{S} spanning all the vertices. This result is asymptotically sharp, and implies in particular that any 3-uniform hypergraph on nn vertices with minimum codegree exceeding n3+o(n)\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