Spanning surfaces in 3-graphs

TL;DR

This paper proves a topological extension of Dirac's theorem for 2-dimensional complexes, showing that high minimum codegree guarantees the existence of spanning surfaces like the sphere, with sharp bounds.

Contribution

It establishes a new topological Dirac-type theorem for 2-complexes, extending classical graph results to higher dimensions and surfaces.

Findings

High minimum codegree ensures spanning surfaces in 3-graphs.

Asymptotically sharp bounds for the existence of such surfaces.

Includes a specific case for spanning triangulations of the 2-sphere.

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.