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.2019.36
URN: urn:nbn:de:0030-drops-104401
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10440/
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Fox, Jacob ; Pach, János ; Suk, Andrew

Semi-Algebraic Colorings of Complete Graphs

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LIPIcs-SoCG-2019-36.pdf (0.4 MB)

Abstract

We consider m-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case m = 2 was first studied by Alon et al., who applied this framework to obtain surprisingly strong Ramsey-type results for intersection graphs of geometric objects and for other graphs arising in computational geometry. Considering larger values of m is relevant, e.g., to problems concerning the number of distinct distances determined by a point set.
For p >= 3 and m >= 2, the classical Ramsey number R(p;m) is the smallest positive integer n such that any m-coloring of the edges of K_n, the complete graph on n vertices, contains a monochromatic K_p. It is a longstanding open problem that goes back to Schur (1916) to decide whether R(p;m)=2^{O(m)}, for a fixed p. We prove that this is true if each color class is defined semi-algebraically with bounded complexity, and that the order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle et al., and on a Szemerédi-type regularity lemma for multicolored semi-algebraic graphs, which is of independent interest. The same technique is used to address the semi-algebraic variant of a more general Ramsey-type problem of Erdös and Shelah.

BibTeX - Entry

@InProceedings{fox_et_al:LIPIcs:2019:10440,
  author =	{Jacob Fox and J{\'a}nos Pach and Andrew Suk},
  title =	{{Semi-Algebraic Colorings of Complete Graphs}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{36:1--36:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Gill Barequet and Yusu Wang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10440},
  URN =		{urn:nbn:de:0030-drops-104401},
  doi =		{10.4230/LIPIcs.SoCG.2019.36},
  annote =	{Keywords: Semi-algebraic graphs, Ramsey theory, regularity lemma}
}

Keywords: Semi-algebraic graphs, Ramsey theory, regularity lemma
Collection: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue Date: 2019
Date of publication: 11.06.2019

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