Hasse diagrams with large chromatic number
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Date
2021-06
Publication Type
Journal Article
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yes
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Abstract
For every positive integer n, we construct a Hasse diagram with n vertices and independence number O(n(3/4)). Such graphs have chromatic number Omega(n(1/4)), which significantly improves the previously best-known constructions of Hasse diagrams having chromatic number Theta(log n). In addition, if we also require girth of at least k >= 5, we construct such Hasse diagrams with independence number at most n(1-1/2k-4+o(1)). The proofs are based on the existence of point-line arrangements in the plane with many incidences and avoids certain forbidden subconfigurations, which we find of independent interest. These results also have the following surprising geometric consequence. They imply the existence of a family C of n curves in the plane such that the disjointness graph G of C is triangle-free (or has high girth), but the chromatic number of G is polynomial in n. Again, the previously best-known construction, due to Pach, Tardos and Toth, had only logarithmic chromatic number.
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published
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Journal / series
Volume
53 (3)
Pages / Article No.
747 - 758
Publisher
Oxford University Press
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Software
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Date created
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Organisational unit
03993 - Sudakov, Benjamin / Sudakov, Benjamin
02500 - Forschungsinstitut für Mathematik / Institute for Mathematical Research
Notes
It was possible to publish this article open access thanks to a Swiss National Licence with the publisher.
Funding
196965 - Problems in Extremal and Probabilistic Combinatorics (SNF)