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Linear Ramsey Numbers

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Combinatorial Algorithms (IWOCA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10979))

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Abstract

The Ramsey number \(R_X(p,q)\) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q. We say that Ramsey number is linear in X if there is a constant k such that \(R_{X}(p,q) \le k(p+q)\) for all p, q. In the present paper we conjecture that Ramsey number is linear in X if and only if the co-chromatic number is bounded in X and determine Ramsey numbers for several classes of graphs that verify the conjecture.

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Acknowledgment

Vadim Lozin acknowledges support from the Russian Science Foundation Grant No. 17-11-01336.

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Authors and Affiliations

  1. Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE, UK

    Aistis Atminas

  2. University of Warwick, Coventry, UK

    Vadim Lozin

  3. Lobachevsky State University of Nizhniy Novgorod, Nizhny Novgorod, Russia

    Vadim Lozin

  4. Department of Computer Science, Durham University, South Road, Durham, DH1 3LE, UK

    Viktor Zamaraev

Authors
  1. Aistis Atminas
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  2. Vadim Lozin
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  3. Viktor Zamaraev
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Corresponding author

Correspondence to Vadim Lozin .

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Editors and Affiliations

  1. Department of Informatics, King’s College London, London, United Kingdom

    Costas Iliopoulos

  2. Department of Computer Science, School of Computing, National University of Singapore, Singapore, Singapore

    Hon Wai Leong

  3. Department of Computer Science, School of Computing, National University of Singapore, Singapore, Singapore

    Wing-Kin Sung

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Atminas, A., Lozin, V., Zamaraev, V. (2018). Linear Ramsey Numbers. In: Iliopoulos, C., Leong, H., Sung, WK. (eds) Combinatorial Algorithms. IWOCA 2018. Lecture Notes in Computer Science(), vol 10979. Springer, Cham. https://doi.org/10.1007/978-3-319-94667-2_3

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  • DOI: https://doi.org/10.1007/978-3-319-94667-2_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-94666-5

  • Online ISBN: 978-3-319-94667-2

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