Spanning surfaces in 3-graphs

Agelos Georgakopoulos; John Haslegrave; Richard Montgomery; Bhargav Narayanan

doi:10.4171/JEMS/1101

Spanning surfaces in 3-graphs

Agelos Georgakopoulos, John Haslegrave, Richard Montgomery, Bhargav Narayanan

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We prove a topological extension of Dirac’s theorem suggested by Gowers in 2005: for any connected, closed surface S, we show that any two-dimensional simplicial complex on n vertices in which each pair of vertices belongs to at least ⁿ₃ C o.n/facets contains a homeomorph of 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 ⁿ₃ C o.n/contains a spanning triangulation of the sphere.

Original language	English (US)
Pages (from-to)	303-339
Number of pages	37
Journal	Journal of the European Mathematical Society
Volume	24
Issue number	1
DOIs	https://doi.org/10.4171/JEMS/1101
State	Published - 2022
Externally published	Yes

All Science Journal Classification (ASJC) codes

Mathematics(all)
Applied Mathematics

Keywords

Dirac’s theorem
Extremal simplicial topology
Spanning structures in hypergraphs
Triangulated surfaces

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10.4171/JEMS/1101

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title = "Spanning surfaces in 3-graphs",

abstract = "We prove a topological extension of Dirac{\textquoteright}s theorem suggested by Gowers in 2005: for any connected, closed surface S, we show that any two-dimensional simplicial complex on n vertices in which each pair of vertices belongs to at least n3 C o.n/facets contains a homeomorph of 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 n3 C o.n/contains a spanning triangulation of the sphere.",

keywords = "Dirac{\textquoteright}s theorem, Extremal simplicial topology, Spanning structures in hypergraphs, Triangulated surfaces",

author = "Agelos Georgakopoulos and John Haslegrave and Richard Montgomery and Bhargav Narayanan",

note = "Funding Information: Funding. The first and second authors were supported by the European Research Council grant 639046 under the European Union{\textquoteright}s Horizon 2020 research and innovation programme. The second author was also partially supported by UK Research and Innovation Future Leaders Fellowship MR/S016325/1. The third author was supported by the European Research Council grant 947978 under the European Union{\textquoteright}s Horizon 2020 research and innovation programme, and the Leverhulme Trust. The fourth author was partially supported by NSF grants CCF-1814409 and DMS-1800521. Publisher Copyright: {\textcopyright} 2021 European Mathematical Society.",

year = "2022",

doi = "10.4171/JEMS/1101",

language = "English (US)",

volume = "24",

pages = "303--339",

journal = "Journal of the European Mathematical Society",

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publisher = "European Mathematical Society Publishing House",

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TY - JOUR

T1 - Spanning surfaces in 3-graphs

AU - Georgakopoulos, Agelos

AU - Haslegrave, John

AU - Montgomery, Richard

AU - Narayanan, Bhargav

N1 - Funding Information: Funding. The first and second authors were supported by the European Research Council grant 639046 under the European Union’s Horizon 2020 research and innovation programme. The second author was also partially supported by UK Research and Innovation Future Leaders Fellowship MR/S016325/1. The third author was supported by the European Research Council grant 947978 under the European Union’s Horizon 2020 research and innovation programme, and the Leverhulme Trust. The fourth author was partially supported by NSF grants CCF-1814409 and DMS-1800521. Publisher Copyright: © 2021 European Mathematical Society.

PY - 2022

Y1 - 2022

N2 - We prove a topological extension of Dirac’s theorem suggested by Gowers in 2005: for any connected, closed surface S, we show that any two-dimensional simplicial complex on n vertices in which each pair of vertices belongs to at least n3 C o.n/facets contains a homeomorph of 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 n3 C o.n/contains a spanning triangulation of the sphere.

AB - We prove a topological extension of Dirac’s theorem suggested by Gowers in 2005: for any connected, closed surface S, we show that any two-dimensional simplicial complex on n vertices in which each pair of vertices belongs to at least n3 C o.n/facets contains a homeomorph of 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 n3 C o.n/contains a spanning triangulation of the sphere.

KW - Dirac’s theorem

KW - Extremal simplicial topology

KW - Spanning structures in hypergraphs

KW - Triangulated surfaces

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EP - 339

JO - Journal of the European Mathematical Society

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IS - 1

ER -

Spanning surfaces in 3-graphs

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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