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Computing the Ramsey number R(4,3,3) using abstraction and symmetry breaking

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Abstract

The number R(4, 3, 3) is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its precise value has remained unknown for almost 50 years. This paper presents a methodology based on abstraction and symmetry breaking that applies to solve hard graph edge-coloring problems. The utility of this methodology is demonstrated by using it to compute the value R(4, 3, 3) = 30. Along the way it is required to first compute the previously unknown set \(\mathcal {R}(3,3,3;13)\) consisting of 78,892 Ramsey colorings.

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Notes

  1. Note that nauty does not directly handle edge colored graphs and weak isomorphism directly. We applied an approach called k-layering described by Derrick Stolee [23].

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Acknowledgments

We thank Stanislaw Radziszowski for his guidance and comments which helped improve the presentation of this paper. In particular Stanislaw proposed to show that our technique is able to find the (4, 3, 3; 29) coloring depicted as Fig. 4.

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Authors and Affiliations

  1. Department of Computer Science, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Michael Codish, Michael Frank & Avraham Itzhakov

  2. School of Computing Science, University of Glasgow, Glasgow, Scotland

    Alice Miller

Authors
  1. Michael Codish
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  2. Michael Frank
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  3. Avraham Itzhakov
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  4. Alice Miller
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Correspondence to Michael Codish.

Additional information

Supported by the Israel Science Foundation, grant 182/13.

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Codish, M., Frank, M., Itzhakov, A. et al. Computing the Ramsey number R(4,3,3) using abstraction and symmetry breaking. Constraints 21, 375–393 (2016). https://doi.org/10.1007/s10601-016-9240-3

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