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DOI: 10.4230/LIPIcs.SoCG.2017.43
URN: urn:nbn:de:0030-drops-72246
URL: http://drops.dagstuhl.de/opus/volltexte/2017/7224/
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Fox, Jacob ; Pach, János ; Suk, Andrew

Erdös-Hajnal Conjecture for Graphs with Bounded VC-Dimension

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LIPIcs-SoCG-2017-43.pdf (0.5 MB)


Abstract

The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e^{(log n)^{1 - o(1)}}. The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, e^{c sqrt{log n}}, due to Erdos and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches the celebrated Erdos-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least e^{Omega(log n)}. Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lovasz-Szegedy and Alon-Fischer-Newman, which is called the "ultra-strong regularity lemma" for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/epsilon)^{O(d)}, improving the original bound of (1/epsilon)^{O(d^2)} in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(n^k)-time algorithm for finding a partition meeting the requirements in the k-uniform setting.

BibTeX - Entry

@InProceedings{fox_et_al:LIPIcs:2017:7224,
  author =	{Jacob Fox and J{\'a}nos Pach and Andrew Suk},
  title =	{{Erd{\"o}s-Hajnal Conjecture for Graphs with Bounded VC-Dimension}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{43:1--43:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Boris Aronov and Matthew J. Katz},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7224},
  URN =		{urn:nbn:de:0030-drops-72246},
  doi =		{10.4230/LIPIcs.SoCG.2017.43},
  annote =	{Keywords: VC-dimension, Ramsey theory, regularity lemma}
}

Keywords: VC-dimension, Ramsey theory, regularity lemma
Seminar: 33rd International Symposium on Computational Geometry (SoCG 2017)
Issue Date: 2017
Date of publication: 08.06.2017


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