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Erdős–Hajnal Conjecture for Graphs with Bounded VC-Dimension

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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 Erdős 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 Erdős–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 Lovász–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 \((1/\varepsilon )^{O(d)}\), improving the original bound of \((1/\varepsilon )^{O(d^2)}\) 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. Finally, we establish tight bounds on Ramsey–Turán numbers for graphs with bounded VC-dimension.

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Notes

  1. A binary semi-algebraic relation E on a point set \(P\subset {{\mathbb {R}}}^d\) is the set of pairs of points (u, v) from P whose 2d coordinates satisfy a boolean combination of a fixed number of polynomial inequalities.

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Acknowledgements

We would like to thank Lisa Sauermann for pointing out a small error in an earlier version of the proof of Lemma 2.3.

Funding

Funding were provided by NSF Division of Mathematical Sciences (Grant Nos. 1800746, 1352121, 1651782), Alfred P. Sloan Foundation, a Packard fellowship, Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (Grant Nos. 200020-144531, 200021-162884, 200021-137574).

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Jacob Fox

  2. Alfréd Rényi Institute, Hungarian Academy of Sciences, 1364, Budapest, Hungary

    János Pach

  3. Department of Mathematics, University of California San Diego, La Jolla, CA, 92093-0112, USA

    Andrew Suk

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  1. Jacob Fox
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  2. János Pach
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Correspondence to Andrew Suk.

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Fox, J., Pach, J. & Suk, A. Erdős–Hajnal Conjecture for Graphs with Bounded VC-Dimension. Discrete Comput Geom 61, 809–829 (2019). https://doi.org/10.1007/s00454-018-0046-5

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