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: https://drops.dagstuhl.de/opus/volltexte/2020/12204/
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Fox, Jacob ; Pach, János ; Suk, Andrew

Bounded VC-Dimension Implies the Schur-Erdős Conjecture

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LIPIcs-SoCG-2020-46.pdf (0.4 MB)


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}
}

Keywords: Ramsey theory, VC-dimension, Multicolor Ramsey numbers
Collection: 36th International Symposium on Computational Geometry (SoCG 2020)
Issue Date: 2020
Date of publication: 08.06.2020


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