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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Nerves, minors, and piercing numbers


Authors: Andreas F. Holmsen, Minki Kim and Seunghun Lee
Journal: Trans. Amer. Math. Soc. 371 (2019), 8755-8779
MSC (2010): Primary 05C10, 05C83, 52A35
DOI: https://doi.org/10.1090/tran/7608
Published electronically: March 19, 2019
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Abstract: We make the first step towards a ``nerve theorem'' for graphs. Let $ G$ be a simple graph and let $ \mathcal {F}$ be a family of induced subgraphs of $ G$ such that the intersection of any members of $ \mathcal {F}$ is either empty or connected. We show that if the nerve complex of $ \mathcal {F}$ has non-vanishing homology in dimension three, then $ G$ contains the complete graph on five vertices as a minor. As a consequence we confirm a conjecture of Goaoc concerning an extension of the planar $ (p,q)$ theorem due to Alon and Kleitman: Let $ \mathcal {F}$ be a finite family of open connected sets in the plane such that the intersection of any members of $ \mathcal {F}$ is either empty or connected. If among any $ p \geq 3$ members of $ \mathcal {F}$ there are some three that intersect, then there is a set of $ C$ points which intersects every member of $ \mathcal {F}$, where $ C$ is a constant depending only on $ p$.


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Additional Information

Andreas F. Holmsen
Affiliation: Department of Mathematical Sciences, KAIST, Daejeon 34141, South Korea
Email: andreash@kaist.edu

Minki Kim
Affiliation: Department of Mathematical Sciences, KAIST, Daejeon 34141, South Korea
Address at time of publication: Department of Mathematics, Technion, Haifa, Israel
Email: kmk90@kaist.ac.kr

Seunghun Lee
Affiliation: Department of Mathematical Sciences, KAIST, Daejeon 34141, South Korea
Email: prosolver@kaist.ac.kr

DOI: https://doi.org/10.1090/tran/7608
Received by editor(s): June 16, 2017
Received by editor(s) in revised form: February 20, 2018
Published electronically: March 19, 2019
Additional Notes: All authors were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03930998).
The first author was also partially supported by Swiss National Science Foundation grants 200020-165977 and 200021-162884.
The second author was partially supported at the Technion by ISF grant no. 2023464 and BSF grant no. 2006099
Minki Kim is the corresponding author
Article copyright: © Copyright 2019 American Mathematical Society