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arXiv · 1808.06864

Spanning surfaces in 3-graphs

Agelos Georgakopoulos·John Haslegrave·Richard Montgomery·Bhargav Narayanan
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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 $n/3 + o(n)$ 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 $n/3+o(n)$ contains a spanning triangulation of the $2$-sphere.

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Agelos Georgakopoulos, John Haslegrave, Richard Montgomery, Bhargav Narayanan. 2018-08-21. Spanning surfaces in 3-graphs. https://doi.org/10.4171/jems%2F1101

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