License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2020.46
URN: urn:nbn:de:0030-drops-122046
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12204/
Fox, Jacob ;
Pach, János ;
Suk, Andrew
Bounded VC-Dimension Implies the Schur-Erdős Conjecture
Abstract
In 1916, Schur introduced the Ramsey number r(3;m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph K_n, there is a monochromatic copy of K₃. He showed that r(3;m) ≤ O(m!), and a simple construction demonstrates that r(3;m) ≥ 2^Ω(m). An old conjecture of Erdős states that r(3;m) = 2^Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.
BibTeX - Entry
@InProceedings{fox_et_al:LIPIcs:2020:12204,
author = {Jacob Fox and J{\'a}nos Pach and Andrew Suk},
title = {{Bounded VC-Dimension Implies the Schur-Erdős Conjecture}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {46:1--46:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Sergio Cabello and Danny Z. Chen},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12204},
URN = {urn:nbn:de:0030-drops-122046},
doi = {10.4230/LIPIcs.SoCG.2020.46},
annote = {Keywords: Ramsey theory, VC-dimension, Multicolor Ramsey numbers}
}
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Keywords: |
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Ramsey theory, VC-dimension, Multicolor Ramsey numbers |
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Collection: |
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36th International Symposium on Computational Geometry (SoCG 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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08.06.2020 |
